Properties of Vectors.

Two vectors \( u = \langle 2, 1 \rangle \) and \( w = \langle 0, 4 \rangle \) are given.
Find a vector v such that u + v = 2w

Use components

\[ \begin{align} u + v &= 2w \\ \end{align} \ \] in components is \[ \begin{align} \langle 2, 1 \rangle + \langle v_1, v_2 \rangle &= \langle 0, 8 \rangle \\ \end{align} \ \]

Solve for v1 and v2

\[ \begin{align} 2 + v_1 &= 0 \\ v_1 &= -2 \end{align} \ \] and \[ \begin{align} 1 + v_2 &= 8 \\ v_2 &= 7 \end{align} \ \] then \[ \begin{align} v &= \langle -2, 7 \rangle \end{align} \ \]

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 \( u = \langle 1, 3 \rangle \) and \( v = \langle 5, 2 \rangle \) are given.
Find a vector w such that u + v + w = 0