Chapter 9, Problem 51RE

To determine

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

Answer to Problem 51RE

The vector $v$ and $w$ are neither parallel and nor orthogonal.

Explanation of Solution

Given information:

The first vector is

  $v=-2i+2j$

The second vector is

  $w=-3i+2j$

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 $\theta$ .

  $\begin{array}{c}\cos \theta =\frac{v·w}{\Vert v\Vert \Vert w\Vert }\\ \cos \theta =\frac{\left(-2i+2j\right)\cdot \left(-3i+2j\right)}{\sqrt{{\left(-2\right)}^2+2^2}\sqrt{{\left(-3\right)}^2+{\left(2\right)}^2}}\\ \cos \theta =\frac{6+4}{\sqrt{8}\sqrt{13}}\\ \cos \theta =\frac{10}{\sqrt{104}}\end{array}$

  $\begin{array}{c}\theta ={\cos }^{-1}\left(\frac{10}{\sqrt{104}}\right)\\ =11.3°\end{array}$

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.

  $\begin{array}{c}v\cdot w=\left(-2i+2j\right)\cdot \left(-3i+2j\right)\\ =6+4\\ =10\end{array}$

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

Therefore, the vector $v$ and $w$ are neither parallel and nor orthogonal.