# What is the projection of < -2, -5, 7> onto < 7,2, -5>?

Apr 10, 2016

$p r o {j}_{\vec{b}} \left(\vec{a}\right) = \frac{59}{78} \left[\begin{matrix}- 7 \\ - 2 \\ 5\end{matrix}\right]$

#### Explanation:

Given : Vectors $\vec{a} = < - 2 , - 5 , 7 >$ and $\vec{b} < 7 , 2 , - 5 >$

Required: Projections of $\vec{a}$ onto $\vec{b}$

Solution Strategy: $p r o {j}_{\vec{b}} \left(\vec{a}\right) = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \cdot \vec{b}$

$\vec{a} \cdot \vec{b} = \left(- 2 \cdot 7\right) + \left(- 5 \cdot 2\right) + \left(7 \cdot \left(- 5\right)\right) = - 59$
$\vec{b} \cdot \vec{b} = 7 \cdot 7 + 2 \cdot 2 + \left(- 5 \cdot - 5\right)$=78

$p r o {j}_{\vec{b}} \left(\vec{a}\right) = \frac{59}{78} \left[\begin{matrix}- 7 \\ - 2 \\ 5\end{matrix}\right]$

Note: the vectors are flattened to 2D in the geometric depiction. You imager simply gives you an idea what a projection is...
Let's see what it looks like geometrically: