# What is the projection of < 2 , -6, 0> onto < -1, -4 , 6 >?

Feb 1, 2017

$= \frac{11}{530} \sqrt{530} < - 1 , - 4 , 6 > = 0.4778 < - 1 , - 4 , 6 >$, nearly..

#### Explanation:

The unit vector in the direction $\vec{b} = < - 1 , - 4 , 6 >$ is

$\frac{1}{|} \vec{b} | \vec{b}$

$= \frac{1}{\sqrt{{\left(- 1\right)}^{2} + {\left(- 4\right)}^{2} + {6}^{2}}} \vec{b}$

$= \frac{1}{\sqrt{53}} \vec{b}$

The projection $\vec{c}$of $\vec{a} = < 2 , - 6 , 0 >$ onto $\vec{b}$ is of

length ( modulus )

$| \vec{a} . \vec{b} \frac{|}{|} \vec{a} | = \frac{22}{\sqrt{40}} = \frac{11}{10} \sqrt{10}$.

Now, the projected vector

$\vec{c} = \frac{11}{10} \sqrt{10}$ (unit vector in the direction $\vec{b}$)

$= \frac{11}{10} \sqrt{10} \left(\frac{1}{\sqrt{53}} < - 1 , - 4 , 6 >\right)$

$= \frac{11}{530} \sqrt{530} < - 1 , - 4 , 6 >$