What is the projection of (3i + 2j - 3k) onto  ( i - j + k)?

Jan 31, 2017

$< - \frac{2}{3} , \frac{2}{3} , - \frac{2}{3} >$

Explanation:

The projection of vector a onto b is...
$\frac{a \cdot b}{|} b {|}^{2} \cdot b$

In this case, a is < 3, 2,-3 > and b is < 1, -1, 1 >. Plug in these vectors and simplify...

$\frac{< 3 , 2 , - 3 > \cdot < 1 , - 1 , 1 >}{|} < 1 , - 1 , 1 > {|}^{2} \cdot b$

$\frac{3 \cdot 1 + 2 \cdot \left(- 1\right) + \left(- 3\right) \cdot 1}{{\left(\sqrt{{1}^{2} + {\left(- 1\right)}^{2} + {1}^{2}}\right)}^{2}} \cdot b$

$\frac{3 - 2 - 3}{\sqrt{3}} ^ 2 \cdot < 1 , - 1 , 1 >$

$- \frac{2}{3} \cdot < 1 , - 1 , 1 >$

$< - \frac{2}{3} , \frac{2}{3} , - \frac{2}{3} >$