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Vector Basics - Example 1

In mathematics, a vector (from the Latin "mover") is a geometric object that has a magnitude (or length) and a direction. Vectors can be added to other vectors according to vector algebra, and can be multiplied by a scalar (real number). Vectors play an important role in physics, especially physics and astronomy, because the velocity and the momentum of an object are expressed by vectors.

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Video Transcript

simple gives us two vectors and we want to find all of these things. So first we want to find the length of you. Well, the length of you remember, it's just gonna be the square root of the sum of the squares of the two components. And so this is going to be the square root of three square sport squared. It's 25 in the square to 25 is five. So the length of us five so next you just want to find U plus V. Okay, But we know how to do that. We just add the components. So we take three and then we're going to subtract one, and then we take four. And then we had one. So we'll get two comma five, right? Next. We want to find U minus V. So this time, instead of adding the opponents were going to subtract the commitment so we're gonna have three minus negative one and then left for minus one That will give us for three. Right? So next we want Thio, take the quantity to times you plus three times of B. And so this time what two vectors? Air Re adding we're adding twice the vector you, but twice the vector you we just double each of the components. So that's six times eight and then three times the components of fee. And then we add this together component wise, and we just get three left and then finally, for the last part, we want the magnitude of U minus two V okay, And this is gonna be the magnitude of Well, I'm gonna take three, four, and I'm going to subtract twice be so that's minus two to Yeah, and this is the magnitude. So I'm just gonna go ahead and do this difference. So that's gonna be three minus negative, too. So that's five and then four minus two to So we just want the magnitude of this vector five to That's the square root of five squared. It's two squared, which is the square root 29. And with these two vectors, U and B, and now noticed, they have three components instead of just two. So these air vectors in space not just in the plane. So first we want the length of you. Well, again, this is just the square root of the sum of squares. The components. So one squared was two squared was two squared is nine. So the length is just the squared nine, which is three. So finding U plus v Well, this is exactly the same as before. We just add the components. So this will be 11 two plus zero, and to one or to to three. Okay, so now for U minus, B, we do the same thing, except we just subtract the components. So it's gonna be one minus one Tu minus zero tu minus one or, in other words, zero to you. What? Okay. And then for to you was three v just take twice. You, too. Four fork. And then at three times of me. So I remember just re scaling a vector. We re scale each component. Well, 30 three. And this is five for seven. Just adding the components. Okay. And then finally, we want the length of you minus to be. So this is the same thing as finding the length of you just want to to and then subtracting twice V, which is to 02 But this is the length of minus one and then to is Europe. But this is just the square root. Once we have any component form is just the square root of the sum of the squares of the components, so minus one squared it's two squared, zero squared, but that's just the square root of sighs.