Chapter 9.5, Problem 10AYU

To determine

To find:

a. The dot product $v\cdot w$.

b. The angle between $v$ and $w$.

c. State whether the vectors are parallel, orthogonal or neither.

Answer to Problem 10AYU

a. 6

b. $18.43°$

c. Neither.

Explanation of Solution

Given:

$v=2i+2j,w=i+2j$

Calculation:

a. $v\cdot w=2\left(1\right)+\left(2\right)2=6$

b. $\Vert v\Vert =\sqrt{{\left(2\right)}^2+{\left(2\right)}^2}=\sqrt{8}=2\sqrt{2}$

$\Vert w\Vert =\sqrt{{\left(1\right)}^2+{\left(2\right)}^2}=\sqrt{5}$

$\cos \theta =\frac{v\cdot w}{\Vert v\Vert \Vert w\Vert }=\frac{6}{2\sqrt{2}\sqrt{5}}=\frac{3}{\sqrt{10}}=\frac{3\sqrt{10}}{10}$

$\theta ={\cos }^{-1}\left(\frac{3\sqrt{10}}{10}\right)=18.43°$

c. The two vectors are neither parallel nor orthogonal to each other.