# What is the projection of (8i + 12j + 14k) onto  (3i – 4j + 4k)?

Jul 29, 2018

The projection is $= \frac{32}{41} \cdot < 3 , - 4 , 4 >$

#### Explanation:

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

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| \vec{a} {|}^{2}} \vec{a}$

Here,

$\vec{a} = < 3 , - 4 , 4 >$

$\vec{b} = < 8 , 12 , 14 >$

Therefore,

The dot product is

$\vec{a} . \vec{b} = < 3 , - 4 , 4 > . < 8 , 12 , 14 > = 24 - 48 + 56 = 32$

The modulus of $\vec{a}$ is

$| \vec{a} | = | < 3 , - 4 , 4 > | = \sqrt{9 + 16 + 16} = \sqrt{41}$

Therefore

$p r o {j}_{\vec{a}} \vec{b} = \frac{32}{41} \cdot < 3 , - 4 , 4 >$