Chapter 9, Problem 87RE

To determine

To calculate:The Dot product $v\cdot w$ and angle between $v$ and $w$ .To use the vectors $v=-2i+j$ and $w=4i-3j$ .

Answer to Problem 87RE

The Dot product $v\cdot w$ and angle between $v$ and $w$ is $-11$ . And the angle between them is $\theta ={\cos }^{-1}\left(\frac{-11}{5\sqrt{5}}\right)$ .

Explanation of Solution

Given information:

The vectors:

  $v=-2i+j$ and $w=4i-3j$

Formula used:

The dot product of the vectors are solved by the formula: $v\cdot w=a_1{a}_2+b_1{b}_2+c_1{c}_2$ .

The formula for finding the angle between two vectors are:

  $\begin{array}{c}\cos \theta =\frac{v\cdot w}{\Vert v\Vert \Vert w\Vert }\\ \theta ={\cos }^{-1}\frac{v\cdot w}{\Vert v\Vert \Vert w\Vert }\end{array}$

The formula to find the magnitude of a vector is as:

  $\begin{array}{l}v=ai+bj+ck\\ \Vert v\Vert =\sqrt{a^2+b^2+c^2}\end{array}$

Calculation:

Consider the vectors:

  $v=-2i+j$ and $w=4i-3j$

First Take a dot product of $v$ and $w$ .

Now apply the Dot product formula of the matrix:

  $\begin{array}{c}v\cdot w=a_1{a}_2+b_1{b}_2+c_1{c}_2\\ v\cdot w=\left(-2i+j\right)\cdot \left(4i-3j\right)\\ =-2\cdot 4+1\cdot -3\\ =-8-3\\ =-11\end{array}$

Now apply the formula to find the angle between the both vectors $v$ and $w$ :

  $\begin{array}{c}\cos \theta =\frac{v\cdot w}{\Vert v\Vert \Vert w\Vert }\\ \theta ={\cos }^{-1}\frac{v\cdot w}{\Vert v\Vert \Vert w\Vert }\end{array}$

Here first find the value of magnitude of $\Vert v\Vert$ .

  $\begin{array}{c}v=-2i+j\\ \Vert v\Vert =\sqrt{{\left(-2\right)}^2+1^2}\\ \Vert v\Vert =\sqrt{5}\end{array}$

Now find the value of magnitude of $\Vert w\Vert$ .

  $\begin{array}{c}w=4i-3j\\ \Vert w\Vert =\sqrt{{\left(4\right)}^2+{\left(-3\right)}^2}\\ \Vert w\Vert =\sqrt{16+9}\\ \Vert w\Vert =\sqrt{25}\\ =5\end{array}$

Now apply the values of magnitude of the both vectors $v$ and $w$ .

Apply the values in the formula:

  $\begin{array}{c}\cos \theta =\frac{v\cdot w}{\Vert v\Vert \Vert w\Vert }\\ =\frac{-11}{\sqrt{5}\cdot 5}\\ =\frac{-11}{5\sqrt{5}}\end{array}$

Thus, The Dot product $v\cdot w$ and angle between $v$ and $w$ is $-11$ . And the angle between them is $\theta ={\cos }^{-1}\left(\frac{-11}{5\sqrt{5}}\right)$ .