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Prove Property 5 of vectors algebraically for the case $ n = 3 $. Then use similar triangles to give a geometric proof.

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$\therefore c(a+b)=c a+c b$

04:40

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 2

Vectors

Campbell University

Harvey Mudd College

Baylor University

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.

00:53

Find all vectors $\mathbf{…

00:27

06:12

Given the three vectors $\…

okay For the human, probably want to prove property five. Which says that if we add the distributive property of vectors, so we're going to have C times A plus B. So let's say we have vectors A and then we have vector B. Well, we know that A plus B is going to be this factor here. So C times A plus B could for example, be a smaller vector. So something like, let's undo that some smaller factor that looks like this, that would be a scalar multiple of it, or it could be some larger vector that looks like this, but regardless, it's going to have that same direction. Just a potentially different magnitude. Now, let's think about the other way. What if we had C. Times those components first? So in this case we're going to have um maybe something like this and then something like this. What we see, we could rearrange this to go right here and we'd end up getting the same factor as a result.

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