The Dot Product and Projections
Part 1 of 6
Consider the vectors u = (2, -2, 2) and v = (2, 5, -2). Find:
i. ||v|| = $\sqrt{2^2 + 5^2 + (-2)^2}$ = $\sqrt{33}$
ii. u \cdot v = 4 - 10 - 4 = -10
Part 2 of 6
It is known that ||u|| = $\sqrt{12}$.
Find the angle between the vectors. Enter an exact value in radians.
$\theta = \arccos(\frac{-10}{\sqrt{33}\sqrt{12}}) = \cos^{-1}(\frac{-5}{3\sqrt{11}})$
Part 3 of 6
Find a unit vector in the direction of v.
$\hat{v} = \langle \frac{2}{\sqrt{33}}, \frac{5}{\sqrt{33}}, \frac{-2}{\sqrt{33}}\rangle$
Part 4 of 6
Find $comp_v u$.
$comp_v u = \frac{-10}{\sqrt{33}} = \frac{-10\sqrt{33}}{33}$
Part 5 of 6
Find $u_\parallel = proj_v u$.
$u_\parallel = proj_v u = \frac{-10}{\sqrt{33}} \langle \frac{2}{\sqrt{33}}, \frac{5}{\sqrt{33}}, \frac{-2}{\sqrt{33}}\rangle$
