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Suppose that $ a $ and $ b $ are nonzero vectors that are not parallel and $ c $ is any vector in the plane determined by $ a $ and $ b $. Give a geometric argument to show that $ c $ can be written as $ c = sa + tb $ for suitable scalars $ s $ and $ t $. Then give an argument using components

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Let $\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle, \mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle$ and $\mathbf{c}=\left\langle c_{1}, c_{2}\right\rangle .$ So $\mathbf{c}= s \mathbf{a}+t \mathbf{b}$ means $\left\langle c_{1}, c_{2}\right\rangle= s\left\langle a_{1}, a_{2}\right\rangle+ t\left\langle b_{1}, b_{2}\right\rangle=$ $\left\langle s a_{1}, s a_{2}\right\rangle+\left\langle t b_{1}, t b_{2}\right\rangle=\left\langle s a_{1}+t b_{1}, s a_{2}+t b_{2}\right\rangle .$ That is $ c_{1}=s a_{1}+t b_{1}$ and $c_{2}=s a_{2}+t b_{2} .$ The argument here is that these two equations form a system of linear equations which has a unique solution for the values of $s$ and $t$ because $\left\langle a_{1}, a_{2}\right\rangle$ and $\left\langle b_{1}, b_{2}\right\rangle$ are not parallel to one another (i.e., they are not multiple of one another).

06:13

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 2

Vectors

Johns Hopkins University

Missouri State University

University of Michigan - Ann Arbor

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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So let's suppose that A. And B are non zero vectors that are not parallel to see and see as any vector in the plane. So we want to give a geometric argument to show that C can be written as um the sum of a plus B. With scalar multiples, S and T. So we can use the argument, give the argument using components. So we know that C is going to be C sub X. He said why? And we want to show that's equal to if we have a plus or a X. A. Y. Name. Plus. Yes, do I? Well we noticed that we could add these apps, so we get A X plus bx and then A Y plus B. Y. But they had scalar multiples. So what we see in this case is that we would have um S A N T. B. So S here T. Here S here T. Here. So when we do this, we see that we can have any scalar multiples to get to be ex well, we know if we just took out the S. And the T. And we take A. X. Plus bx and multiplied by some scalar multiple, then that's going to end up giving us C. X. And the same thing holds for the white components, so we see that this is going to hold.

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