Chapter 6, Problem 42RE

(a)

To determine

Theresultant vector $u+v$ and plot.

Answer to Problem 42RE

Theresultant vector $u+v=\langle 4,4\rangle$

Explanation of Solution

Given info:

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

Formula used:

A vector whose initial point is the origin $(0,0)$ can be uniquely represented by thecoordinates of its terminal point $(v_1,v_2)$ . This is the component form of a vector v,written as $v=\langle v_1,v_2\rangle$

Calculation:

We have

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

So,

  $\begin{array}{l}u+v=\langle 4,5\rangle +\langle 0,-1\rangle \\ u+v=\langle 4,4\rangle \end{array}$

Theresultant vector $u+v=\langle 4,4\rangle$

Plot is given by

  

Conclusion:

Thus,theresultant vector $u+v=\langle 4,4\rangle$

(b)

To determine

Theresultant vector $u-v$ and plot.

Answer to Problem 42RE

Theresultant vector $u-v=\langle 4,6\rangle$

Explanation of Solution

Given info:

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

Formula used:

A vector whose initial point is the origin $(0,0)$ can be uniquely represented by thecoordinates of its terminal point $(v_1,v_2)$ . This is the component form of a vector v,written as $v=\langle v_1,v_2\rangle$

Calculation:

We have

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

So,

  $\begin{array}{l}u-v=\langle 4,5\rangle -\langle 0,-1\rangle \\ u-v=\langle 4,6\rangle \end{array}$

Theresultant vector $u-v=\langle 4,6\rangle$

Plot is given by

  

Conclusion:

Thus,theresultant vector $u-v=\langle 4,6\rangle$

(c)

To determine

Theresultant vector $4u$ and plot.

Answer to Problem 42RE

Theresultant vector $4u=\langle 16,20\rangle$

Explanation of Solution

Given info:

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

Formula used:

A vector whose initial point is the origin $(0,0)$ can be uniquely represented by thecoordinates of its terminal point $(v_1,v_2)$ . This is the component form of a vector v,written as $v=\langle v_1,v_2\rangle$

Calculation:

We have

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

So,

  $\begin{array}{l}4u=4\langle 4,5\rangle \\ 4u=\langle 16,20\rangle \end{array}$

Theresultant vector $4u=\langle 16,20\rangle$

Plot is given by

  

Conclusion:

Thus,theresultant vector $4u=\langle 16,20\rangle$

(d)

To determine

Theresultant vector $3v+5u$ and plot.

Answer to Problem 42RE

Theresultant vector $3v+5u=\langle 20,22\rangle$

Explanation of Solution

Given info:

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

Formula used:

A vector whose initial point is the origin $(0,0)$ can be uniquely represented by thecoordinates of its terminal point $(v_1,v_2)$ . This is the component form of a vector v,written as $v=\langle v_1,v_2\rangle$

Calculation:

We have

  $\begin{array}{l}u=\langle 4,5\rangle \\ v=\langle 0,-1\rangle \end{array}$

So,

  $\begin{array}{l}3v+5u=3\langle 0,-1\rangle +5\langle 4,5\rangle \\ 3v+5u=\langle 20,22\rangle \end{array}$

Theresultant vector $3v+5u=\langle 20,22\rangle$

Plot is given by

  

Conclusion:

Thus,theresultant vector $3v+5u=\langle 20,22\rangle$