Chapter 9, Problem 50RE

To determine

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

Answer to Problem 50RE

The vector $v$ and $w$ are parallel.

Explanation of Solution

Given information:

The first vector is

  $v=2i+3j$

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 $\theta$ .

  $\begin{array}{c}\cos \theta =\frac{v·w}{\Vert v\Vert \Vert w\Vert }\\ \cos \theta =\frac{\left(2i+3j\right)\cdot \left(-4i-6j\right)}{\sqrt{2^2+3^2}\sqrt{{\left(-4\right)}^2+{\left(-6\right)}^2}}\\ \cos \theta =\frac{-8-18}{\sqrt{13}\sqrt{52}}\\ \cos \theta =-1\end{array}$

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

Since, the angle between $v$ and $w$ is $\pi$ . Hence, the vectors are 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.

  $\begin{array}{c}v\cdot w=\left(2i+3j\right)\cdot \left(-4i-6j\right)\\ =-8-18\\ =-26\end{array}$

Since, the dot product is not equal to zero hence the vectors are not orthogonal.

Therefore, the vector $v$ and $w$ are parallel.