What is the projection of  (4 i + 4 j + 2 k) onto (- 5 i + 4 j - 5 k)?

Dec 27, 2017

The projection is $= - \frac{7}{33} < - 5 , 4 , - 5 >$

Explanation:

The vector projection of $\vec{b}$ onto $\vec{a}$

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| | \vec{a} | |} \vec{a}$

Here,

$\vec{b} = < 4 , 4 , 2 >$

$\vec{a} = < - 5 , 4 , - 5 >$

The dot product is

$\vec{a} . \vec{b} = < 4 , 4 , 2 > . < - 5 , 4 , - 5 > = \left(4 \cdot - 5\right) + \left(4 \cdot 4\right) + \left(2 \cdot - 5\right) = - 20 + 16 - 10 = - 14$

The modulus of $\vec{b}$ is

$| | \vec{a} | | = \sqrt{{\left(- 5\right)}^{2} + {\left(4\right)}^{2} + {\left(- 5\right)}^{2}} = \sqrt{66}$

Therefore,

$p r o {j}_{\vec{a}} \vec{b} = \frac{- 14}{66} \cdot < - 5 , 4 , - 5 >$

$= - \frac{7}{33} < - 5 , 4 , - 5 >$