91Ó°ÊÓ

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ |\mathbf{v}| \mathbf{u}+|\mathbf{u}| \mathbf{v} $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ \frac{1}{2} \mathbf{u} \cdot \mathbf{v} $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ (\mathbf{v} \cdot \mathbf{w}) \mathbf{u} $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ (\mathbf{u}+\mathbf{v}) \cdot \mathbf{w} $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ (\mathbf{u}-\mathbf{v}) \cdot(\mathbf{v}+\mathbf{w}) $$

Find a unit vector in the opposite direction of \(\mathbf{v}=\) \(\langle 10,-5,10\rangle .\)

Find a unit vector in the same direction as \(\mathbf{v}=\mathbf{i}-\) \(3 \mathbf{j}+2 \mathbf{k}\)

Find a vector \(\mathbf{u}\) that is four times as long as \(\mathbf{v}=\mathbf{i}\) \(-\mathbf{j}+\mathbf{k}\) in the same direction as \(\mathbf{v}\).

Use (6) to find the cross product of the given vectors. $$ \mathbf{u}=\langle 1,-3,1\rangle, \mathbf{v}=\langle 2,0,4\rangle $$

Use (6) to find the cross product of the given vectors. $$ \mathbf{u}=2 \mathbf{i}+\mathbf{j}-\mathbf{k}, \mathbf{v}=-6 \mathbf{i}-3 \mathbf{j}+3 \mathbf{k} $$