WEBVTT mathematics/linear-algebra/hovasapian
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Welcome back to educator.com and welcome back to linear algebra, this is going to be lesson number 11, and we are going to talk about N vectors today.
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In the last lesson we talked about vectors in the plane, which are two vectors, because each vector is represented by two numbers, so when we talk about a vector in free space, we just, it's a vector, it's a, it's called a 3 vector.
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We also call it R₃, which will symbolize in just a little bit, when we speak about n vectors, it's just any number of them, if we are talking about a 10 dimensional space, it just means a 1 by 10 matrix, or a 10 by 1 matrix you remember.
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It's just 0 numbers, so that’s the nice thing about mathematics is you are not, you are not tied to what you can represent as far as reality is concerned, is just as real, and but you know obviously we don't know how to draw 10 space or an space or 13 space.
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But these things do exist and mathematics is actually exactly the same, so let's get started....
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... Okay, so let's just throw out a few examples, a four vector...
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... Something like (1, 3, -2, 6), again we are just talking about a 4 by 1; I could also have written this vector as (1, 3, -2, 6).
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It really doesn't matter, later on it will make a difference depending on how we want to arrange it, because we are going to be multiplying these things by matrices, so sometimes we want it this way, sometimes we want it this way.
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Another representation is just regular coordinate representation, X, Y, Z, so on, so I could also write this as (1, 3, -2, 6).
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They all mean the same thing, it just depends on what it is that you are doing, okay, a seven vector, let's do...
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... Now again, so you would have (0, 5, 0, 6, 9, 7, 2), something like that.
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And again you can write it as a row, you can write it as a list in coordinate form, this just means that you have this many dimensional space.
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Two dimensional space is two numbers, three dimensional space is 3 numbers, this is a seven dimensional space, perfectly valid, perfectly real.
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And the mathematics is handled exactly the same way as it was last times, okay let's talk about vector addition.
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last time we talked about vector addition, when we said will you add two vectors together, you are just adding the individual components, in other words the individual numbers of those vectors together.
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Lets just do an example, so let's say...
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... The vector addition, let's call our, let's do a vector, so we have (1, -2 and 3), and let's take B as (2, 3 and -3), so when we add them we are just adding the 1 and the 2, the -2 and the 3, the 3 and the -3, and so our U + V...
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... Equals...
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... Again it's a, it's a three vector, 1 + 2 is 3, -2 + 3 is -1, I am sorry, +1.
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And 3 - is 0, so we have the vector (3, 1, 0)...
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.. If we do scalar multiplication, when we take a vector and when we just multiply by a scalar, I will write with our number...
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... If I wanted to do, let's say, let's use U again, if I wanted to do 5U, well I just multiply everything in there by 5, 5 times 1 is 5, 5 times -2, -10, 5 times 3 is 15.
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5U equals, and it's again, it's a three vector, nothing changes, we have (5, -10 and 15), so vector addition, scalar multiplication just like when we did it for vectors in the plane, you just have more numbers.
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Okay, now let us write down the theorem, this is going to be a little bit of writing and again a lot of this you have already seen before, but it's sort of nice to write it over and over again, because it solidifies it in your mind.
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And it's nice to see it formally in a mathematical sense, again in mathematics we try to be as precise as possible to leave no room for error, so let's write this one out, and there one aspect of this theorem that I am going to digress on it.
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It's going to be a very important aspect and you will see it in just a minute, so we will let U, V, W...
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... Be N vectors, or vectors in N space, we also talk about at that way, so 3 vector is our vectors in 3 space.
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And space, and just to let you know the symbol for let's say R₃ is R here with the double line, it stands for the real numbers.
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And we put a little 3 up there, it means that we are taking one number from each real number line, if you think of 3 space, you think of it as in Z axis and X axis and a Y axis.
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Each one of those axes represents the real number line, so since we are using three real number lines that are mutually perpendicular to each other, that's where this 3 comes from.
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If we talk about N space, it's symbolized R < font size="-6" > n < /font > .
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Okay, so let's put a little 1 here and we will start with an A, so if we have U, V and W as N vectors, U + V...
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... Is a vector in N space, you know this already. If I take two vectors to three vectors and if I add them together, I get a three vector, so it's not like I land in some other space.
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I start with 3, I do something to them called addition, and I actually, the result that I get is still a three vector.
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Now, this might seem natural to you, but as it turns out it's not quite so natural, there are things, situations, mathematical structures where this is not true, and this is the digression that I am going to go on in a moment.
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When we say that, so U + V is a vector in N space, this property is called closure, okay, and we say that vector addition is closed...
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... Under addition...
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... We talk about the property of closure, or we also say that this operation of addition of vector is closed under addition, and here is what that means that, and again you might think that this is perfectly natural.
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Why would it be anything else, for example if I take the numbers 5 and 6 and if I add them together, I get 11, which is just another number.
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In other words I still end up with a number, well consider this, let's just take the set of odd numbers, so (1, 3, 5, 7, 9) etc.
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And let's take the set of even numbers, so all I have done is I split the number system and actual number system into even and odd, so this odd, this is even,
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Now, let's just start with some even numbers, if I take any two even numbers, and let's just take the number 4 and the number 8, and if I add them together, so I perform an operation with two elements of that set 8 and 4, I end up with 12.
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But 12 is an even number, so an even + an eve, I end up with an even number.
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But what that means is that I start with two things, I do something to it, but I end up back in my same set, I don't leave, this is called closure.
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That means I don't land someplace else, but now try this with the odd numbers, so let's take an odd number like 3 and let's add it to another odd number let's say 5.
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But when I add these together, I get 8, I get an even number, so I start with two elements in this set, I do something to them, I add them and yet all of a sudden I end up in a different space.
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I have separated these two, why is it that the even numbers when you add to elements, you don't leave that space, you still end up with an even number, but now if I add two odd numbers, I end up outside of that space.
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I didn't end up with an odd number, this is not odd, so as it turns out, this property of closure is actually a very deep property, and we have to specify it.
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As it turns out, when n you add two vectors together, two N vectors, you get an N vector, but this example, this counter example is, it demonstrates that it doesn't always have to be the case.
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that's why it's important for us to specify it, so something that seems obvious in mathematics usually has a very deep reason underlying it, that's why we say this.
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Okay, let's continue, B, U + V = V + U, that means you can ad in either order, so vector addition is commutative.
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C, U + V + W = exactly what you think, U + V quantity + W, so vector addition is associative, okay.
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D, there exist a unique, remember this symbol, reverse E means there exists a unique, that's what that little exclamations, that means there is 1, only 1.
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When we say there exists, it could be more than one, but when we say there exists a unique, we are making a very specific statement that it's the only one that exists.
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There exist a 0 vector...
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... Such that U + the 0 vector equals the 0 vector + U commutativity, where you get U back, that is called the additive identity, identity meaning you start with the vector.
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You add something to it, nothing changes you get that vector back, it's an identity.
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And last but not least, okay...
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... For each U, there exists a unique vector symbolized -U, such that U + this -U vector...
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... Gives me the 0 vector, this is called the additive inverse, again 5 + -5 gives u 0, 5 + 0 gives you 5, so this 0 vector is called the additive identity.
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And additive because it's specific to this property of addition, doesn't apply to multiplication, we will get to multiplication in just a little bit, and for each U for every vector in N space, there exists a vector -U, such that when you add them together, you get a 0 vector, so they come in pairs okay...
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Now the second part, which actually deals with scalar multiplication, if I take some constant times U, well that's also closed.
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As it turns out, if I have a vector in N space and if I multiply it by some scalar, I end up with a vector in N space, I don't jump to another space.
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Again it seems natural, it's obvious, you have been doing it all your life, but there is something deeper going on, it doesn't have to be that way., and as you saw an example of something that you deal with every day, the odd numbers.
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The odd numbers don't satisfy this property, the odd numbers are not closed under addition when you add two odds together, you end up outside of the set, you end up with an even not an odd.
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We have to specify closure; it’s a very important property...
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We say this is closed under scalar multiplication, whereas before it was closed under vector addition, okay.
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C times U + V = C times U + C times V, so the distributive property under scalar multiplication is active, I can distribute the scalars over the vectors that I add excuse me.
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If I add two scalars together and multiply them by some vector, well I can distribute the scalar, the vector over the scalars.
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Its C, I guess I chose a V here, I meant to do a U, but that's fine, it doesn't really matter + D times V, okay.
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And C times D times U = CD times U, so if I have some vector or I multiply uit by a number, then I multiply it by another one.
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I can take a vector and I can just multiply the two numbers together and then multiply it by the vector, and again all these are very common properties that you are accustomed to.
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And 1 times U = U, this 1 again we are talking about scalar multiplication here, scalar...
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... Scalar multiply, this is just the number 1, it is not the vector 1, it is not the unit vector that we talked about before, it is the number 1 times U, gives me U back.
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This is called the multiplicative identity, it is the element that when I multiply by the vector, it gives me back the vector, nothing changes, before we talked about the additive identity, the 0, so that when I added it to a vector, I got the vector back.
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Okay, let's talk a little bit about coordinate systems, as it turns out, there are two types of coordinate systems, there's something called a right handed and a left handed, generally unless there is a reason for doing so.
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It is just been conventional in mathematics to use a right handed coordinate system, and will show what it is that, that means here, we will draw both of them so that you know.
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Z...
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... Okay, there is going to be times when I forget my arrows, forgive me I sometimes just don't write my arrows, Z, Y and X, okay...
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... This is a right handed system, what you don't notice, X is, so the Z and the Y are actually in the plane of paper.
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And it's as if we draw this going back also, the X axis is the one that's out, coming out towards you and away from you, this is the right handed system.
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And the reason it's called right handed because if we actually take our right hand and sort of make a little L with this like this.
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Some people do it with finger like this, I don't know, I think it's a little less intuitive, just sort of keep your hand at an angle like that, your arm is, end up being the Y axis, your thumb is the Z axis, and your fingers are the X axis, what we would consider like the primaries.
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Once we establish our fingers moving in the direction of X axis, the Z and the Y sort of take this particular shape, left handed would be the other way around, and we will draw that, just so you see what it looks like.
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And this is of ‘course in R₃, so in 3 space because we can actually represent it, as you know we can't represent 4 dimensional, 5 dimensional or other spaces.
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R₃ is where we have the right handed an left handed systems, okay w so we have that goes there...
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That's that, that's that, and all you have done with the left handed system is switched X and Y, so X and Z are in the plane, and Y is the thing that comes forward and away from you.
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And again, X is always your fingers, X Y, Z, if you arrange like this, you will actually see this is the left handed system, but again we are considered with the right handed system.
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This is what we are going to be dealing with primarily, right handed..
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... The Z and the Y are in the plane, it's the x that's coming towards you, it's as if we have taken the X, Y plane that we are used to and we have flipped it forward toward you, and now that X is pointing towards you, and the Z is up.
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Okay...
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... Let's talk about the...
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... Projection of a point onto a coordinate plane, very important operation...
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... Projection of a point...
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... Onto a...
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... Coordinate line or plane actually, because I am, on my first example I am going to do is going to be a two dimensional example, so that you can see it, and then we will do the three dimensional.
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When we talk about R₂, two space, we have our X, Y coordinate, this is X, this is Y...
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... That's fine, I don't need to label them, let's say we pick a point right there, and let's label that point, let's say it's the point (3, 4), so 3 in the X direction, 4 in the Y direction.
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When I project something onto one of the coordinate axes, it's as if I am shining a light down that way, and what I do is I drop a perpendicular from that point onto that axis.
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I end up at the point 3, if I project this way, I project onto the Y axis, I end up here because that's all you are doing with projections, is you are starting with your point and you are going down to one of your axes or to the plane.
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And you are literally sort of dropping off that whatever coordinates you are not talking about, so if I project onto the X axes, I drop a perpendicular onto the X axis and where I end up, this is my projection right here.
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Now let's do it for 3 space...
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Okay, when I draw the vector, it's going to seem a little strange, but once I do the projection, it will e very clear what's happening, so these are little label, this is Y, this is X, this is Z.
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I have a vector, okay, let's say that the vector is (2, 3, 4)...
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... Now, I want to project this onto the XY plane, in other words I just want to shine a light on it, and cause, I want to see the shadow, that's what the projection is, it's a shadow on the particular X, Y plane.
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I am going to shine a light on it from above, which means I drop a perpendicular...
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... Down to...
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... The X, Y, and now from the origin, I draw that point, and because I dropped it down to the X, Y plane, this point is (2, 3).
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Now we are in the X, Y plane, the shadow of this vector on the X, Y plane is that, that is the actual shadow of that thing, makes complete sense.
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You can project it onto the ZY plane, you can project it onto the ZX plane, in fact let's do that.
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If I project it onto the ZX plane, I would have something like....
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... Something like that, and you might have perpendicular that way, and then you would have a vector in the ZX plane, and that would be, so you take the Z and the X.
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It would be (2, 4), 2 in this direction, 4 in this direction, because now I ignored the Y.
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Here I have projected it, cast a shadow onto the XY axis, XY, which means I only take the XY, so I have a vector in the XY plane, which is the shadow, which is the projection of this.
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Projections are going to be very important in linear algebra, because again any time you drop a perpendicular to something, you are talking about the shortest distance something.
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The shortest distance between this point and this point is that length; we will talk more about that in a little bit.
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Okay, let's move on...
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... Let U be an N vector, so now we are not specifying the space, we are just saying generally speaking, the magnitude of U is exactly what you think it is.
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You just take, all of the, oh, let's actually list this, so it will be something like U₁, U₂, so on all the way to U < font size="-6" > n < /font > right, that many entries, so it is going to be Un² + U2², + so on and so on.
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Until Un² all of it under the radical, this is just a Pythagorean theorem in N dimensions.
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The Pythagorean was just for...
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... U₁ and U₂, in 3 space, it's U₁, U₂, U₃, in 15 space it's U₁, U₂, U₃, all the way to U < font size="-6" > 15 < /font > , mathematics is handled exactly the same way.
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Square each entry, add them all together, take the square root, perfectly valid...
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Okay, let's see, let's also define the distance between two points...
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... Well points are nothing but vectors, so we can speak of them as points or we can speak of them as vectors, which is an arrow from the origin to that point, so the distance between two points.
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And I know you have seen the distance formula before, the distance between two points and vector form is the magnitude...
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... If one of the points, so if point 1, is represented by some vector U, and point 2 is some vector V, the distance between them is the magnitude of the vector U - V.
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In other words take U, subtract v, you will still get an N vector and then take the magnitude of that, meaning apply this, well, when you write it out, you get, well U - V in component form is (U₁ - V₁)².
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+ (U₂ - V₂)² + so on all the way to (U < font size="-6" > n < /font > - V < font size="-6" > n < /font > )² and all of this under the radical sign.
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I prefer to use a radical sign instead of putting parenthesis and doing to this, to the power of one half, just a personal preference.
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Symbolism is I mean it's important, but ultimately it's about your understanding, so...
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... Okay, so you see that everything that we discussed in lesson 2 for 2 space is exactly the same, it's completely analogous, you just have more coordinates, more numbers to deal with, that's the only thing that's different.
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And if you remember we also had another...
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... Way of representing the magnitude in terms of A, the vector itself, if we took U, dotted it with itself and to of the square root, that's also another way of finding the magnitude.
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Okay, now we are going to discuss a very important inequality in mathematics, well it's profoundly important inequality, it's called the quotient words in equality.
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And it might similarly strange but, that actually does make sense, so put an arrow there...
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... Quotient words...
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... Excuse me...
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... Real briefly I just want to speak really generally about inequalities, because in a minute we are going to introduce the second inequality called the triangular inequality and there are many inequalities in mathematics.
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In the branch of mathematics that most of you know is calculus, most mathematicians refer to it as analysis, and analysis is exactly what you think, it's any other kind of analysis.
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You have a certain amount of data in front of you, and you are trying to come up to come up with some sort of conclusion for what that data is implying.
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Well often you don't really have all of the information at your disposal, so you have to analysis the situation, you are basically breaking it up seeing what you do have, and seeing how the pieces fit together.
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Well as it turns out, really what you are doing is an analysis as you are establishing relationships between the bits of information that you have at your disposal, and relationships, one relation is an equality relation.
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But in analysis often, you can speculate about the equality of something, but you can say something about the inequality between those two things.
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So as it turns out in mathematical analysis in equalities play a central role because they allow us to order things in a certain way, and extract information that way.
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Analysis really is about dealing with inequalities, relationships among bits of information, so the quotient words in equality says, if I have two vectors U and V...
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... Let's write that out actually, let U and V be numbers of R and, in other words N vectors, you know...
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... Then, excuse me, the absolute value of U.V...
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... Is less than or equal to...
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... The magnitude of U...
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... times the magnitude of V, okay so let's stop and think about this for a second, let's make sure that our symbolism is understood.
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A vector with thee two double line, that's magnitude, that's the length of the vector, these are numbers, so if I take the, this single one here, it's absolute value.
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Now, U.V is a number, it's a scalar, so you can take the absolute value of a, of a number sometimes you doubt V will be negative, sometimes it will be positive.
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That's why these absolute value signs are here, well magnitudes are always positive, so we don't have to worry about absolutes here.
00:27:56.000 --> 00:28:10.000
Well magnitudes are always positive, so we don't have to worry about absolutes here, what this says, if I have two random vectors, and if I take the dot products of those vectors.
00:28:10.000 --> 00:28:21.000
the absolute value of the dot product is always going to be less than or equal to the product of the magnitudes, what this is saying, it's placing an upper limit on what the dot product can be.
00:28:21.000 --> 00:28:36.000
That's profoundly important, we need to know that, we are not just, you know the number is not just going off to infinity, so it actually plays an upper limit on what this dot product can be, it's going to turn out to be very important.
00:28:36.000 --> 00:28:47.000
I'll go ahead and give you an informal justification for this, as supposed to an actual proof, I want you take this informal justification as , not with the grain of salt, but take it lightly.
00:28:47.000 --> 00:28:57.000
This is not a proof; in general, what we do is we end up proving it and we end up using the quotient words in equality to...
00:28:57.000 --> 00:29:07.000
... I am doing something a little backwards, we use the quotient words in equality once we have proved it, to go through this justification to define the angle between two vectors.
00:29:07.000 --> 00:29:22.000
Now we did that one, we were working in R₂, we just gave the definition, however I am going to use that in order to sort of justify that this is possible, just to sort of let you know that it is possible, this doesn't just drop out of the sky, okay.
00:29:22.000 --> 00:29:35.000
Remember last lesson, we said that the cosine of an angle between the two vectors is equal to...
00:29:35.000 --> 00:29:45.000
... IU.V divided by the magnitude of U times, the magnitude of V.
00:29:45.000 --> 00:29:53.000
Well, the cosine of an angle is always...
00:29:53.000 --> 00:30:10.000
Is between -1 and +1, that's you know this, the cosine curve, goes like that, the +1 is in upper limit, -1 is in lower limit, therefore when I have this -1 and +1, I can actually write this, this way.
00:30:10.000 --> 00:30:21.000
And say if the cosine of θ it's absolute value sign just allows me to not write it this way a little short hand, okay.
00:30:21.000 --> 00:30:39.000
Equal to, so they take the absolute value of that, well that's the absolute value of this whole thing, and since the bottom is positive, it doesn't really matter, I don't need the absolute value sign there, okay.
00:30:39.000 --> 00:30:48.000
Magnitude of U times the magnitude of V is less than or equal to 1, so now I have this thing, okay.
00:30:48.000 --> 00:30:58.000
Start off with the definition that I had, I noted it's between -1 and +1, the cosine θ that means that this value is between -1, and +1.
00:30:58.000 --> 00:31:06.000
Take the absolute value of the cosine so that I can illuminate this and just write it this way, well the absolute value of the cosine is the absolute value of this thing.
00:31:06.000 --> 00:31:22.000
Now I have that, now just multiply through and what you get is...
00:31:22.000 --> 00:31:32.000
... Absolute value of the dot product of two vectors is less, I should probably make this a little more clear here, this is a little odd, is less than or equal to the product of their magnitudes.
00:31:32.000 --> 00:31:49.000
This is the quotient words in equality, and this always holds, so again this is just an informal justification to let you know that this is, this sort of make sense based on what you know about cosine θ, we actually do the other way around.
00:31:49.000 --> 00:31:54.000
Okay, let's move forward...
00:31:54.000 --> 00:32:00.000
... Just do a little example here, so...
00:32:00.000 --> 00:32:16.000
... We said an this also holds for N vectors, the cosine θ is U.V divided by the magnitude of U, writing all these out is exhausting.
00:32:16.000 --> 00:32:38.000
Okay, so we will let U = (1, 0, 0, 1), that's our for vector U, and we will let V = (0, 1, 0, 1) okay...
00:32:38.000 --> 00:32:48.000
... U.V, this times that + this times that + this times that + this times that, all of these are 0, so U.V = 1.
00:32:48.000 --> 00:32:53.000
Magnitude of U...
00:32:53.000 --> 00:33:03.000
...This squared + that square + his squared + that squared, (0, 0), these are (1, 1), square root, radical 2.
00:33:03.000 --> 00:33:18.000
The magnitude of V, same thing, that squared + that squared + that squared + that squared, under the radical sign we get rad 2, therefore we have using this formula, just putting them in.
00:33:18.000 --> 00:33:42.000
Cosine of θ equals 1, over the magnitude of 1 times magnitude of the other, radical 2 times radical 2, we end up with 1 half, cosine θ equals 1 half, and if you remember your trigonometry, θ equals the inverse cosine of 1 half, that is going to be a 60 degree angle, for in terms of radians π over 6.
00:33:42.000 --> 00:33:55.000
If I have this vector (1, 0, 0, 1) and I have the vector (0, 1, 0, 1), I know that the angle between them is 60 degrees, π over 6 radians.
00:33:55.000 --> 00:34:15.000
Okay, now another property U.V equals 0 if and only if, meaning it's equivalent to U and V or...
00:34:15.000 --> 00:34:19.000
... Orthogonal...
00:34:19.000 --> 00:34:29.000
... In two space and in three space, orthogonal is the same as perpendicular, but when we are dealing with N vectors, we don't really have a way of visualizing, let's say 13 space.
00:34:29.000 --> 00:34:40.000
But we know if 13 vector exists, we can write it, we can do the math word, it's a very real thing, so we don't use the term perpendicular, because that's more geometric as far as the real world is concerned.
00:34:40.000 --> 00:34:56.000
We use the term orthogonal, so orthogonal is a generalized term for perpendicular, so U.V is 0, that means that U and V are orthogonal, this if and only if means well if U and V are orthogonal, then I know that U.V equals 0.
00:34:56.000 --> 00:35:09.000
The implication goes in both directions, that’s all this if and only if means, you can also write this three lines, there is an equivalence, this is the same as that, either one is fine, you can replace this with this, this with this.
00:35:09.000 --> 00:35:18.000
Okay, U.V...
00:35:18.000 --> 00:35:25.000
... When the absolute value of U.V actually equals, when there is a strict equality of the magnitudes...
00:35:25.000 --> 00:35:31.000
... Of the magnitudes...
00:35:31.000 --> 00:35:45.000
... If and only if U and V are parallel, well which makes sense, if you have U, this way and if you have V this way.
00:35:45.000 --> 00:35:55.000
Well the angle between them is a 180 degrees, they are parallel, or the other possibility is U that way and V that way, if they are in the same direction, the angle between them.
00:35:55.000 --> 00:36:06.000
If I put them right on top of each other is 0, well the cosine of θ is a cosine of 0 is 1, the cosine of a 180- degrees is -1.
00:36:06.000 --> 00:36:18.000
that's where this inequality in the quotient works in equality becomes the strict equality, so if I take the dot product and they are equal to 0, I know that they are orthogonal, perpendicular.
00:36:18.000 --> 00:36:27.000
If I take the dot product and they happen to equal, the product of the magnitudes, I know that they are actually parallel.
00:36:27.000 --> 00:36:44.000
Okay, now let's introduce the other inequality that we talked about, this is called the triangle inequality, also a very important inequality, and this one is very intuitive, because there is a picture for it that you can add, that makes sense.
00:36:44.000 --> 00:36:58.000
In fact, those of you that remember from algebra 1 and 2, you probably spent about half a day deciding whether certain triangle when they give you the length of sides is possible, you were doing the triangle inequalities, what you were doing.
00:36:58.000 --> 00:37:06.000
The triangle inequality says I'll do the algebra, then I'll do the picture, I want to do it the other way around, we are dealing with linear algebra, to deal algebraically.
00:37:06.000 --> 00:37:15.000
Pictures will help us, but it's not, pictures are not proof, you know we want to become a custom to actually letting the algebra do the work for us.
00:37:15.000 --> 00:37:19.000
Say's that the magnitude...
00:37:19.000 --> 00:37:34.000
... And again U and V are N vectors, the magnitude of U + V, once I add U + V,. is less than or equal to the magnitude of U + the magnitude of V.
00:37:34.000 --> 00:37:48.000
It places in upper limit on the sum of two vectors, the sum of two vectors, the biggest, that the sum of two vectors can ever be is the length of one vector + the length of the other vector.
00:37:48.000 --> 00:37:53.000
This is a inequality, here's what this means, and this is why it's called the triangle inequality.
00:37:53.000 --> 00:37:57.000
Let's draw two vectors...
00:37:57.000 --> 00:38:01.000
... Let's say I have vector U...
00:38:01.000 --> 00:38:15.000
... And I will label vector U, and let's say I have vector V, notice that I didn't draw them from any, you know any frame of reference, or just random vectors...
00:38:15.000 --> 00:38:33.000
... Adding vectors means you start with one, and then wherever you end up like for example you start with 1, and then wherever you end up, you add the other one, you just put it on top of it and you go to that one, the point where you end up from your original starting point to your final ending point.
00:38:33.000 --> 00:38:47.000
That's your vector addition, it just means add them in order, do U first, then do V, so in this case, let's do U, it's here, and here, and then we will do V, which is here.
00:38:47.000 --> 00:39:00.000
Okay, son that’s V, we end up here, this vector right here is our U + V, notice what this says.
00:39:00.000 --> 00:39:12.000
it says that the length of this vector U + V is less than or equal to the length of this + the length of this, all that means is that the third side of the triangle is less than or equal to the sum of the two sides.
00:39:12.000 --> 00:39:35.000
that's all that means, because if it were longer, then the sum of the two sides, what you would get is triangle like that, let's say that's one side, let's say that's another side, let's say that's another side, the triangle doesn't close.
00:39:35.000 --> 00:39:38.000
These will just collapse onto there, in order to have a triangle; the sum of the two side, of sum of any two sides has to be at least bigger than, has to be bigger than the third side.
00:39:38.000 --> 00:39:45.000
Any time it's equal, well that situation is just when...
00:39:45.000 --> 00:39:53.000
... They basically lay on top of each other, what you have is a line, these collapses...
00:39:53.000 --> 00:39:59.000
... Precisely to align, so again in order to have a triangle, there actually has to be a strict inequality, that's all it means.
00:39:59.000 --> 00:40:07.000
Once again, the sum of two sides of a triangle is always going to be bigger than the third side, that's the only way a triangle can actually exist, that's why it's called the triangle inequality.
00:40:07.000 --> 00:40:18.000
As it turns out, it has nothing to do with the pictures, just because we can draw a picture, and we call this thing a triangle, this is an algebraic property, this is true in any number of dimensions.
00:40:18.000 --> 00:40:35.000
And in fact it has absolutely nothing to do with a picture, pictures are our representations of making things clear, this is a deep mathematical algebraic property, okay...
00:40:35.000 --> 00:40:39.000
... Unit vectors...
00:40:39.000 --> 00:40:51.000
... Again a unit vector is just a vector, with a length of 1...
00:40:51.000 --> 00:41:05.000
... And our symbol, my symbol for that is just X unit, what you do is you take the particular vector you are dealing with X, and you multiply it by the reciprocal of its magnitude, that's it.
00:41:05.000 --> 00:41:21.000
All you are doing is taking the vector, dividing it by its length, just like when you take a number 10, divided by 10 you get 1., well you can't divide by a vector, but you can divide by the magnitude of the vector, because the magnitude is a number, okay...
00:41:21.000 --> 00:41:32.000
... In the last section we introduced the vector I, and the vector J, they were unit vectors in the X direction...
00:41:32.000 --> 00:41:43.000
... X direction and a unit vector in the Y direction, now we are going to introduce the unit vector K, it is a unit vector in the Z direction.
00:41:43.000 --> 00:41:50.000
Let me draw my right handed coordinate system again...
00:41:50.000 --> 00:42:03.000
... Let me darken this up that is a vector of length 1 that is I in the X direction that is a vector in the Y direction, called J.
00:42:03.000 --> 00:42:17.000
And the U vector of length 1 that moves in the Z direction is called K...
00:42:17.000 --> 00:42:20.000
... Any...
00:42:20.000 --> 00:42:29.000
... Vector in R₃, R₃, 3 vector, 3 space...
00:42:29.000 --> 00:42:35.000
... Can be represented...
00:42:35.000 --> 00:42:39.000
... As a...
00:42:39.000 --> 00:42:43.000
... Linear...
00:42:43.000 --> 00:42:46.000
... Combination...
00:42:46.000 --> 00:42:49.000
... Of...
00:42:49.000 --> 00:43:02.000
... The vectors I, j and K, in other words I can take any vector and I can actually write it as a sum, that's what linear combination means, you are just adding.
00:43:02.000 --> 00:43:16.000
Of these unit vectors, very important unit vectors, very important unit vectors, so for example if I had...
00:43:16.000 --> 00:43:35.000
... U = (0, 4, 2, 3), now let's say I have V is equal to (0, -1, 2, 0)...
00:43:35.000 --> 00:43:52.000
... I might say I can write U as 0I + 4J, actually excuse me, let's forget, these are, we are dealing with 3 vector not 4, + 2K.
00:43:52.000 --> 00:44:03.000
All I have done is I have taken this vector and I have represented it, that means I move 0 in this direction, I move 4 in this direction, and I move 2 in this direction.
00:44:03.000 --> 00:44:18.000
And that's all it is, that's all these unit vectors do, they are a sort of A for a more reference, that allows any vector in R₂ or R₃ to be represented as a linear combination as sum of these vectors.
00:44:18.000 --> 00:44:25.000
We will get a little bit more into this later, when we actually break things up, okay.
00:44:25.000 --> 00:44:35.000
Now let's see what we have got, okay so let's do a little but of a recap and we will finish off with some examples...
00:44:35.000 --> 00:44:54.000
... Orthogonal vectors, these are the important points, orthogonal vectors, when U.V is equal to 0, if and only if U and V are...
00:44:54.000 --> 00:45:04.000
... Orthogonal, or ortho, so really important, orthogonal vectors, perpendicular vectors, or when the dot product of those vectors equals 0 and the other way around.
00:45:04.000 --> 00:45:18.000
Quotient works in equality, very important in equality, it says that the absolute value of U.V less than or equal to the magnitude of U times...
00:45:18.000 --> 00:45:35.000
... the magnitude of , profoundly important in equality, triangle inequality says that the magnitude of the sum of U and V is less than or equal to the magnitude of U +...
00:45:35.000 --> 00:45:40.000
... The magnitude of V...
00:45:40.000 --> 00:45:52.000
... Okay, let's do some examples here, let's let U = what it is we had before, so (0, 4, 2 and 3).
00:45:52.000 --> 00:46:09.000
We will let V = (0, -1, 2, 0) and I wrote it in coordinate form, makes no difference, let's calculate U.V.
00:46:09.000 --> 00:46:27.000
U.V is you multiply, so 0 times 0 is 0, 4 times -1, -4, 2 times 2 +4, 3 times 0, 0, 0 - 4 + 4 = 0 = 0.
00:46:27.000 --> 00:46:39.000
Dot product is 0, so U and V are orthogonal...
00:46:39.000 --> 00:46:57.000
... let's find a unit vector in the direction of U, okay, so we are looking for U unit, well, I know that, that's equal to 1 over the magnitude of U times U itself.
00:46:57.000 --> 00:47:02.000
1 over the magnitude, that's just a scalar, by multiplying the scalar by the vector.
00:47:02.000 --> 00:47:13.000
Okay, let's see what the magnitude of U is, magnitude of U equals...
00:47:13.000 --> 00:47:27.000
... 0 + 16, 2 times 2 is 4, 3 times 3 is 9, all under the radical sign...
00:47:27.000 --> 00:47:42.000
... Radical 29, therefore our unit vector is 1 over radical 29 times...
00:47:42.000 --> 00:47:55.000
... (0, 4, 2, 3) it's equal to (0, 4, over radical 29.
00:47:55.000 --> 00:48:13.000
2 over radical 29, and 3 over radical 29, this is 4 vector, but now this vector has a length of 1, if you were to find the magnitude of this vector, it would be 1, it's in the direction of U, but it has a length of 1.
00:48:13.000 --> 00:48:25.000
Alright, okay let's do one final example, a little bit more complex, sort of tie and some other things that we did in previous lessons.
00:48:25.000 --> 00:48:30.000
We want to find...
00:48:30.000 --> 00:48:40.000
... A vector V, which is (A, B, C), such that...
00:48:40.000 --> 00:48:46.000
... V is ortho...
00:48:46.000 --> 00:49:00.000
... To both W, which is (1, 2, 1) and X, which is equal to (1, -1, 1).
00:49:00.000 --> 00:49:16.000
Okay, so we have the vectors W, and we have the vector W and X, and we want to find a vector (A, B, C), in other words we want to find (A, B, C), at least 1, does that for all of them, but at least 1, such that V is orthogonal to both.
00:49:16.000 --> 00:49:31.000
Well we know what orthogonal is, orthogonal means that V.W is 0, and V.X is also 0, so this is use that definition, write out some equations and see what we get, so...
00:49:31.000 --> 00:49:43.000
... V.W is A times 1 is A, + B times 2 is 2B + C, and we know that that's equal to 0, that's all I have done here.
00:49:43.000 --> 00:50:00.000
I have used the definition of dot product and I have written out a linear equation, A + 2B + c = 0, well V.X, I also know that it's equal to 0, well so V it's just A times 1 is A.
00:50:00.000 --> 00:50:18.000
A times -1 is -B, and C times 1 is C, that's equal to 0, well I have two equations, three unknowns, let's go ahead and subject this to reduced row echelon, the Gauss Jordan elimination, and let's see what we can do.
00:50:18.000 --> 00:50:29.000
let's form our augmented matrix here, so (1, 2, 1, 0), (1, 2, 1), let me put the whole thing there so we know that we are dealing with the 0's over here.
00:50:29.000 --> 00:50:35.000
And (1, -1, 1)...
00:50:35.000 --> 00:50:55.000
... We are going to subject this to reduced row echelon form, and when we do that, we end up with the following, we end up with (1, 0, 1, 0, 0, 1, 0, 0).
00:50:55.000 --> 00:51:07.000
This is reduced row echelon, this first column is A, the second column is B, this one is fine, this one is fine, this one, this is not, there is no leading entry here it's free.
00:51:07.000 --> 00:51:19.000
As it turns out, this third variable C can be absolutely anything, therefore our solution is the following, C = anything...
00:51:19.000 --> 00:51:27.000
... Well B = 0...
00:51:27.000 --> 00:51:46.000
... That's what this does, it allows us to just read off what's there, so B + 0 = 0, so B = 0, and now A is equal to well actually let me write it differently A + C = 0.
00:51:46.000 --> 00:52:04.000
Therefore A equals -C, or negative anything, because i can choose anything for my C, so let's just say that ZC = 5, that means B = 0, and A = -5.
00:52:04.000 --> 00:52:14.000
One possible answer is (-5, 0, 5) for my vector V...
00:52:14.000 --> 00:52:23.000
... This vector is orthogonal to that vector and that vector, and again C can be anything, so it's not the only vector.
00:52:23.000 --> 00:52:28.000
There is a whole sleeve of vectors, it's an infinite number of them, so this has an infinite number of solutions.
00:52:28.000 --> 00:52:38.000
And what we did is we just used the definition of dot product, and we use the fact that we know that any time two vectors when they are dotted and equals 0, they are orthogonal to each other.
00:52:38.000 --> 00:52:44.000
Okay, thank you for joining us here at educator.com, linear algebra, we will see you next time.