Chapter 9, Problem 52RE

To determine

To find: Whether $v$ and $w$ are parallel, orthogonal, or neither.

Answer to Problem 52RE

The vectors $v$ and $w$ are orthogonal.

Explanation of Solution

Given information:

The first vector is

  $v=3i-2j$

The second vector is

  $w=4i+6j$

Calculation:

The two vectors $v$ and $w$ are known as to be parallel if there is a nonzero scalar $\alpha$ so that $v=\alpha w$ . The angle between $v$ and $w$ is $0$ or $\pi$ .

Calculate the angle $\theta$ .

  $\begin{array}{c}\cos \theta =\frac{v·w}{\Vert v\Vert \Vert w\Vert }\\ \cos \theta =\frac{\left(3i-2j\right)\cdot \left(4i+6j\right)}{\sqrt{{\left(3\right)}^2+{\left(-2\right)}^2}\sqrt{{\left(4\right)}^2+{\left(6\right)}^2}}\\ \cos \theta =\frac{12+12}{\sqrt{13}\sqrt{52}}\\ \cos \theta =0\end{array}$

  $\begin{array}{c}\theta ={\cos }^{-1}\left(0\right)\\ =\frac{\pi }{2}\end{array}$

Since, the angle between $v$ and $w$ is not $0$ or $\pi$ . Hence, the vectors are not parallel.

If the angle between two vectors $v$ and $w$ is $\frac{\pi }{2}$ then the vectors are called orthogonal, that is $v\cdot w=0$

Calculate the dot product of two vectors.

  $\begin{array}{c}v\cdot w=\left(3i-2j\right)\cdot \left(4i+6j\right)\\ =12-12\\ =0\end{array}$

Since, the dot product is equal to zero hence the vectors are orthogonal.

Therefore, the vectors $v$ and $w$ are orthogonal.