Chapter 14.2, Problem 17WE

a.

To determine

To find: The vector w in component form.

Answer to Problem 17WE

The vector w in component form is $6i+7j$ .

Explanation of Solution

Given:

The following vectors:

  $\begin{array}{c}u=4i+3j\\ v=2i-j\\ w=2u-v\end{array}$

Calculation:

Substitute the values of vectors u and v in the expression $2u-v$ to determine the value of vector w.

  $\begin{array}{c}w=2u-v\\ =2\left(4i+3j\right)-\left(2i-j\right)\\ =\left(8-2\right)i+\left(6+1\right)j\\ =6i+7j\end{array}$

So, vector w in component form is $6i+7j$ .

b.

To determine

To draw: A diagram showing the vectors u, v and w.

Answer to Problem 17WE

Figure 1 represents the diagram showing the vectors u, v and w.

Explanation of Solution

Given:

The following vectors:

  $\begin{array}{c}u=4i+3j\\ v=2i-j\\ w=2u-v\end{array}$

Calculation:

The diagram showing the vectors u, v and w is given below:

  

Figure 1

c.

To determine

To find: The value of $\Vert w\Vert$ .

Answer to Problem 17WE

The value of $\Vert w\Vert$ is $\sqrt{85}$ .

Explanation of Solution

Given:

The vector $w=6i+7j$

Calculation:

For a vector $z=ai+bj$ , the value of $\Vert z\Vert$ is given by the formula $\Vert z\Vert =\sqrt{a^2+b^2}$ .

Substitute 6 for a and 7 for b in the formula $\Vert w\Vert =\sqrt{a^2+b^2}$ to determine the value of $\Vert w\Vert$ .

  $\begin{array}{c}\Vert w\Vert =\sqrt{6^2+7^2}\\ =\sqrt{36+49}\\ =\sqrt{85}\end{array}$

d.

To determine

To find: The angle $\gamma$ , $0°\le \gamma \le 360°$ , that w makes with the positive x -axis.

Answer to Problem 17WE

The angle $\gamma$ is $49.41°$ .

Explanation of Solution

Given:

The vector $w=6i+7j$

Calculation:

For a vector $z=ai+bj$ , the angle $\gamma$ that vector z makes with the positive x -axis is calculated using the formula $\gamma ={\tan }^{-1}\frac{b}{a}$ .

Substitute 6 for a and 7 for b in the formula $\gamma ={\tan }^{-1}\frac{b}{a}$ to determine the value of $\gamma$ .

  $\begin{array}{c}\gamma ={\tan }^{-1}\frac{7}{6}\\ \approx 49.41°\end{array}$