Chapter 6, Problem 79CR

To determine

To calculate:

The following $u$ and $v$ are parallel, orthogonal or neither:

  $\begin{array}{l}u=\langle 8,5\rangle \\ v=\langle -2,4\rangle \end{array}$

Answer to Problem 79CR

The vectors $u$ and $v$ are neither orthogonal nor parallel

Explanation of Solution

Given information:

  $\begin{array}{l}u=\langle 8,5\rangle \\ v=\langle -2,4\rangle \end{array}$

Formula used:

The angle between two vectors $u$ and $v$ is given by the following expression:

  $\cos \theta =\frac{u\cdot v}{\Vert u\Vert \Vert v\Vert }$

Calculation:

Know that:

  $\begin{array}{l}u=\langle 8,5\rangle \\ v=\langle -2,4\rangle \end{array}$

The dot product of given two vectors $u$ and $v$ is calculated as:

  $\begin{array}{l}u\cdot v=u_1{v}_1+u_2{v}_2\\ u\cdot v=\left(8\right)\left(-2\right)+\left(5\right)\left(4\right)\\ u\cdot v=-16+20\\ u\cdot v=4\end{array}$

Now, the magnitude of vector $u$ and $v$ is calculated as:

  $\begin{array}{l}\Vert u\Vert =\sqrt{u_1{}^2+u_2{}^2}=\sqrt{{\left(8\right)}^2+{\left(5\right)}^2}=\sqrt{89}\\ \Vert v\Vert =\sqrt{v_1{}^2+v_2{}^2}=\sqrt{{\left(-2\right)}^2+{\left(4\right)}^2}=2\sqrt{5}\end{array}$

Now, the angle between two vectors $u$ and $v$ is calculated as:

  $\begin{array}{l}\cos \theta =\frac{u\cdot v}{\Vert u\Vert \Vert v\Vert }\\ \cos \theta =\frac{4}{\left(\sqrt{89}\right)\left(2\sqrt{5}\right)}\\ \theta ={\cos }^{-1}\left(\frac{2}{\sqrt{445}}\right)\\ \theta =1.4758\end{array}$

So, the angle between two vectors $u$ and $v$ is $\theta =1.4758$ which shows that two vectors are neither orthogonal nor parallel.