Chapter 9.5, Problem 8AYU

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 8AYU

Solution:

a. 0

b. ${90}^{°}$

c. Orthogonal.

Explanation of Solution

Given:

  $v=i+j,w=-i+j$

Calculation:

a. $v\cdot w=1(-1)+1(1)=0$

b. $\Vert v\Vert =\sqrt{{(1)}^2+{(1)}^2}=\sqrt{2}$

  $\Vert w\Vert =\sqrt{{(-1)}^2+{(1)}^2}=\sqrt{2}$

  $\cos \theta =\frac{v\cdot w}{\Vert v\Vert \Vert w\Vert }=\frac{0}{\sqrt{2}\sqrt{2}}=0$

  $\theta ={\cos }^{-1}(0)={90}^{°}$

c. The two vectors are orthogonal to each other, since $v\cdot w=0$ .