What is the projection of < 5 ,- 3, 9 > onto < 8 , 8 , -4 >?

Dec 15, 2017

The projection is $= - \frac{5}{3} < 8 , 8 , - 4 >$

Explanation:

The vector projection of $\vec{v}$ onto $\vec{u}$ is

$p r o {j}_{u} v = \frac{\vec{u} . \vec{v}}{| | \vec{u} | |} ^ 2 \cdot \vec{u}$

Here,

$\vec{v} = < 5 , - 3 , 9 >$ and

$\vec{u} = < 8 , 8 , - 4 >$

The dot product is

$\vec{u} . \vec{v} = < 5 , - 3 , 9 > . < 8 , 8 , - 4 >$

$= \left(5 \cdot 8\right) + \left(- 3 \cdot 8\right) + \left(9 \cdot - 4\right) = 40 - 24 - 36 = - 20$

The magnitude of $\vec{u}$ is

$= | | < 8 , 8 , - 4 > | | = \sqrt{{\left(8\right)}^{2} + {\left(8\right)}^{2} + {\left(- 4\right)}^{2}}$

$= \sqrt{64 + 64 + 16} = \sqrt{144} = 12$

The vector projection is

$p r o {j}_{u} v = - \frac{20}{12} \cdot < 8 , 8 , - 4 > = - \frac{5}{3} < 8 , 8 , - 4 >$