What is the projection of <8,-5,3 > onto <7,6,0 >?

Feb 5, 2018

The projection is $= \frac{26}{85} < 7 , 6 , 0 >$

Explanation:

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

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

$\vec{a} = < 7 , 6 , 0 >$

$\vec{b} = < 8 , - 5 , 3 >$

The dot product is

$\vec{a} . \vec{b} = < 7 , 6 , 0 > . < 8 , - 5 , 3 > = \left(7\right) \cdot \left(8\right) + \left(6\right) \cdot \left(- 5\right) + \left(0\right) \cdot \left(3\right)$

$= 56 - 30 + 0 = 26$

The modulus of $\vec{a}$ is

$| | \vec{a} | | = | | < 7 , 6 , 0 > | | = \sqrt{{\left(7\right)}^{2} + {\left(6\right)}^{2} + {\left(0\right)}^{2}}$

$= \sqrt{49 + 36 + 0} = \sqrt{85}$

Therefore,

$p r o {j}_{\vec{a}} \vec{b} = \frac{26}{\sqrt{85}} ^ 2 \cdot < 7 , 6 , 0 >$

$= \frac{26}{85} < 7 , 6 , 0 >$