# Given the vectors u=<<2,2>>, v=<<-3,4>>, and w=<<1,-2>>, how do you find (3w*v)u?

Feb 8, 2017

$\left(3 w \cdot v\right) \vec{u} = \left\langle- 66 , - 66\right\rangle$

#### Explanation:

Inner Product Definition
If $\underline{u} = \left\langle\left({u}_{1} , {u}_{2}\right)\right\rangle$, and $\underline{v} = \left\langle\left({v}_{1} , {v}_{2}\right)\right\rangle$, then the inner product (or dot product), a scaler quantity, is given by:

$\underline{u} \cdot \underline{v} = {u}_{1} {v}_{1} + {u}_{2} {v}_{2}$

So, $\vec{u} = \left\langle2 , 2\right\rangle$, $\vec{v} = \left\langle- 3 , 4\right\rangle$, and $\vec{w} = \left\langle1 , - 2\right\rangle$

Then the inner product $\left(3 w \cdot v\right)$ is given by;

$3 \vec{w} \cdot \vec{v} = 3 \left\langle1 , - 2\right\rangle \cdot \left\langle- 3 , 4\right\rangle$
$\text{ } = \left\langle3 , - 6\right\rangle \cdot \left\langle- 3 , 4\right\rangle$
$\text{ } = \left(3\right) \left(- 3\right) + \left(- 6\right) \left(4\right)$
$\text{ } = - 9 - 24$
$\text{ } = - 33$

And so the vector $\left(3 w \cdot v\right) \vec{u}$ is:

$\left(3 w \cdot v\right) \vec{u} = - 17 \vec{u}$
$\text{ } = - 33 \left\langle2 , 2\right\rangle$
$\text{ } = \left\langle- 66 , - 66\right\rangle$