Let ${\bf a} = \langle -6, 2, -7 \rangle$ and ${\bf b} = \langle 6, 10, 1\rangle$ be vectors. Find the scalars and vectors defined below. Note that these formulas only depend on ${\hat {\bf a}} = \frac{{\bf a}}{\| {\bf a} \|}$, the unit vector in the direction of $\bf a$.

The component of $\bf b$ along $\bf a$:
${\rm comp}_{\bf a} {\bf b} = {\hat {\bf a}}\cdot {\bf b} =$

The projection of $\bf b$ onto $\bf a$: (Note: you can write this without square roots.)
${\rm P}_{\bf a} {\bf b} = ({\hat {\bf a}}\cdot {\bf b}) {\hat {\bf a}} = \langle$ , , $\rangle$

The projection of $\bf b$ orthogonal to $\bf a$: (This is defined in terms of the previous projection.)
${\rm P}_{\bf a}^{\perp} {\bf b} = {\bf b} - {\rm P}_{\bf a} {\bf b} = \langle$ , , $\rangle$

Hint: The meaning of these projections is seen from examples where $\bf a = {\bf i} = \langle 1, 0, 0 \rangle$.

${\rm comp}_{\bf i} \langle 2, 1, 5 \rangle = 2$,

${\rm P}_{\bf i} \langle 2, 1, 5 \rangle = \langle 2, 0, 0 \rangle$, and

${\rm P}_{\bf i}^\perp \langle 2, 1, 5 \rangle = \langle 0, 1, 5 \rangle$