# How do you find the scalar and vector projections of b onto a given a = ‹3, 2, 1›, b = ‹0, 1, 3›?

Nov 21, 2015

$p r o {j}_{a} \left(b\right) = \left(0 , \frac{1}{2} , \frac{3}{2}\right)$

#### Explanation:

$p r o {j}_{a} \left(b\right) = v = \lambda b$

$v + n = a R i g h t a r r o w n = a - \lambda b$

$n \cdot b = 0$

$\left(a - \lambda b\right) \cdot b = 0$

$a \cdot b = \lambda b \cdot b$

$\lambda = \frac{a \cdot b}{b \cdot b} = \frac{0 + 2 + 3}{0 + 1 + 9} = \frac{5}{10} = \frac{1}{2}$

$v = \frac{1}{2} b = \left(0 , \frac{1}{2} , \frac{3}{2}\right)$