Question

Find vector z, given u=<-1,3,2>, v=<1,-2,-2> and w=<5,0,-5>
z= -7u+v-(1/5)w

Answer 1

I write my vectors vertically because it's clearer to read. They mean the same thing.
\(\vec u =\begin{bmatrix} -1 \\ 3 \\ 2\end{bmatrix}\), \(\vec v =\begin{bmatrix} -1 \\ 2 \\ -2\end{bmatrix}\), \(\vec w =\begin{bmatrix} 5 \\ 0 \\ -5\end{bmatrix}\)
So, following the formula:
\(\vec z= -7\begin{bmatrix} -1 \\ 3 \\ 2\end{bmatrix} + \begin{bmatrix} -1 \\ 2 \\ -2\end{bmatrix} - \dfrac{1}{5}\begin{bmatrix} 5 \\ 0 \\ -5\end{bmatrix}\)
Use scalar multiplication of vectors.
\(\vec z= \begin{bmatrix} -7 \cdot -1 \\-7 \cdot 3 \\-7 \cdot 2\end{bmatrix} + \begin{bmatrix} -1 \\ 2 \\ -2\end{bmatrix} + \begin{bmatrix} -\frac{1}{5}\cdot 5 \\ -\frac{1}{5}\cdot 0 \\ -\frac{1}{5}\cdot -5\end{bmatrix}\)
\(\vec z= \begin{bmatrix} 7 \\-21 \\-14\end{bmatrix} + \begin{bmatrix} -1 \\ 2 \\ -2\end{bmatrix} + \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}\)
Can you take it from here? Add like components together (vector addition).

Answer 2

Whoops...I copied \(\vec v\) down slightly wrong...but I'm sure you get the idea.