Operations with Vectors.

 

Two vectors \( v = \langle -2, 5 \rangle \) and \( w = \langle 3, 4 \rangle \) are given. Find
(a) 1/3 w
(b) v + 2w
(c) w - v

The scalar multiple of a vector is the scalar multiple of its components.

\[ \begin{align} \frac{1}{3} w &= \frac{1}{3}\langle 3, 4 \rangle \\ &= \langle 1, 4/3 \rangle \\ \end{align} \ \]

Add the components of the two vectors

\[ \begin{align} v + 2w &= \langle -2, 5 \rangle + 2\langle 3, 4 \rangle \\ &= \langle -2, 5 \rangle + \langle 6, 8 \rangle \\ &= \langle -2+6, 5+8 \rangle \\ &= \langle 4, 13 \rangle \end{align} \ \]

Subtract the components of the second from the first

\[ \begin{align} w - v &= \langle 3, 4 \rangle - \langle -2, 5 \rangle \\ &= \langle 5, -1 \rangle \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

Two vectors \( v = \langle -1, 4 \rangle \) and \( w = \langle 2, 5 \rangle \) are given. find
(a) 2/3 w
(b) 2v + w
(c) w - 1/2v