Chapter 3, Problem 3.23P

(a)

To determine

The addition of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

Answer to Problem 3.23P

The addition of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $2\widehat{\text{i}}-6\widehat{\text{j}}$ .

Explanation of Solution

Given info: The value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $3\widehat{\text{i}}-2\widehat{\text{j}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ respectively.

Write the expression for addition of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

${\overrightarrow{\text{R}}}_1=\overrightarrow{\text{A}}+\overrightarrow{\text{B}}$ (1)

Substitute $3\widehat{\text{i}}-2\widehat{\text{j}}$ for $\overrightarrow{\text{A}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ for $\overrightarrow{\text{B}}$ in equation (1) to get value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

$\begin{array}{c}{\overrightarrow{\text{R}}}_1=\left(3\widehat{\text{i}}-2\widehat{\text{j}}\right)+\left(-\widehat{\text{i}}-4\widehat{\text{j}}\right)\\ =\left(3\widehat{\text{i}}-\widehat{\text{i}}\right)+\left(-2\widehat{\text{j}}-4\widehat{\text{j}}\right)\\ =\left(\text{2}\widehat{\text{i}}\right)+\left(-6\widehat{\text{j}}\right)\\ =\text{2}\widehat{\text{i}}-6\widehat{\text{j}}\end{array}$

Conclusion:

Therefore, the addition of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $2\widehat{\text{i}}-6\widehat{\text{j}}$.

(b)

To determine

The subtraction of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

Answer to Problem 3.23P

The subtraction of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $\text{4}\widehat{\text{i}}+2\widehat{\text{j}}$.

Explanation of Solution

Given info: The value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $3\widehat{\text{i}}-2\widehat{\text{j}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ respectively.

Write the expression for subtraction of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

${\overrightarrow{\text{R}}}_2=\overrightarrow{\text{A}}-\overrightarrow{\text{B}}$ (2)

Substitute $3\widehat{\text{i}}-2\widehat{\text{j}}$ for $\overrightarrow{\text{A}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ for $\overrightarrow{\text{B}}$ in equation (2) to get value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

$\begin{array}{c}{\overrightarrow{\text{R}}}_2=\left(3\widehat{\text{i}}-2\widehat{\text{j}}\right)-\left(-\widehat{\text{i}}-4\widehat{\text{j}}\right)\\ =\left(3\widehat{\text{i}}+\widehat{\text{i}}\right)+\left(-2\widehat{\text{j}}+4\widehat{\text{j}}\right)\\ =\left(\text{4}\widehat{\text{i}}\right)+\left(\text{2}\widehat{\text{j}}\right)\\ =4\widehat{\text{i}}+\text{2}\widehat{\text{j}}\end{array}$

Conclusion:

Therefore, the subtraction of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $\text{4}\widehat{\text{i}}+2\widehat{\text{j}}$.

(c)

To determine

The magnitude of addition of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

Answer to Problem 3.23P

The magnitude of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $6.32$.

Explanation of Solution

Given info: The value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $3\widehat{\text{i}}-2\widehat{\text{j}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ respectively.

Write the expression for addition of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

${\overrightarrow{\text{R}}}_1=\overrightarrow{\text{A}}+\overrightarrow{\text{B}}$ (3)

Substitute $3\widehat{\text{i}}-2\widehat{\text{j}}$ for $\overrightarrow{\text{A}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ for $\overrightarrow{\text{B}}$ in equation (3) to get value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

$\begin{array}{c}{\overrightarrow{\text{R}}}_1=\left(3\widehat{\text{i}}-2\widehat{\text{j}}\right)+\left(-\widehat{\text{i}}-4\widehat{\text{j}}\right)\\ =\left(3\widehat{\text{i}}-\widehat{\text{i}}\right)+\left(-2\widehat{\text{j}}-4\widehat{\text{j}}\right)\\ =\left(\text{2}\widehat{\text{i}}\right)+\left(-6\widehat{\text{j}}\right)\\ =\text{2}\widehat{\text{i}}-6\widehat{\text{j}}\end{array}$

Write the expression for the magnitude of two vectors $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is,

$\left|{\overrightarrow{\text{R}}}_1\right|=\sqrt{{\left({\text{R}}_1\right)}_{\text{x}}{}^2+{\left({\text{R}}_1\right)}_{\text{y}}{}^2}$ (4)

Here,

${\left({\text{R}}_1\right)}_{\text{x}}$ is $x$-axis coordinate.

${\left({\text{R}}_1\right)}_{\text{y}}$ is $y$-axis coordinate.

Substitute $2$ for ${\left({\text{R}}_1\right)}_{\text{x}}$ and $-6$ for ${\left({\text{R}}_1\right)}_{\text{y}}$ in equation (4) to get magnitude of two

Vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

$\begin{array}{c}\left|{\overrightarrow{\text{R}}}_1\right|=\sqrt{{\left(2\right)}^2+{\left(-6\right)}^2}\\ =\sqrt{4+36}\\ =\sqrt{40}\\ =6.32\end{array}$

Conclusion:

Therefore, the magnitude of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $6.32$.

(d)

To determine

The magnitude of subtraction of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

Answer to Problem 3.23P

The magnitude of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $4.47$.

Explanation of Solution

Given info: The value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $3\widehat{\text{i}}-2\widehat{\text{j}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ respectively

Write the expression for subtraction of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

${\overrightarrow{\text{R}}}_2=\overrightarrow{\text{A}}-\overrightarrow{\text{B}}$ (5)

Substitute $3\widehat{\text{i}}-2\widehat{\text{j}}$ for $\overrightarrow{\text{A}}$ and $-\widehat{\text{i}}-4\widehat{\text{j}}$ for $\overrightarrow{\text{B}}$ in equation (5) to get value of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

$\begin{array}{c}{\overrightarrow{\text{R}}}_2=\left(3\widehat{\text{i}}-2\widehat{\text{j}}\right)-\left(-\widehat{\text{i}}-4\widehat{\text{j}}\right)\\ =\left(3\widehat{\text{i}}+\widehat{\text{i}}\right)+\left(-2\widehat{\text{j}}+4\widehat{\text{j}}\right)\\ =\left(\text{4}\widehat{\text{i}}\right)+\left(\text{2}\widehat{\text{j}}\right)\\ =4\widehat{\text{i}}+\text{2}\widehat{\text{j}}\end{array}$

Write the expression for the magnitude of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is,

$\left|{\overrightarrow{\text{R}}}_2\right|=\sqrt{{\left({\text{R}}_2\right)}_{\text{x}}{}^2+{\left({\text{R}}_2\right)}_{\text{y}}{}^2}$ (6)

Here,

${\left({\text{R}}_2\right)}_{\text{x}}$ is $x$-axis coordinate.

${\left({\text{R}}_2\right)}_{\text{y}}$ is $y$-axis coordinate.

Substitute $4$ for ${\left({\text{R}}_2\right)}_{\text{x}}$ and $2$ for ${\left({\text{R}}_2\right)}_{\text{y}}$ in equation (6) to get magnitude of two

Vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$.

$\begin{array}{c}\left|{\overrightarrow{\text{R}}}_2\right|=\sqrt{{\left(4\right)}^2+{\left(2\right)}^2}\\ =\sqrt{16+4}\\ =\sqrt{20}\\ =4.47\end{array}$

Conclusion:

Therefore, the magnitude of two vector $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is $4.47$.

(e)

To determine

The direction of vector $\left|{\overrightarrow{\text{R}}}_1\right|$ and $\left|{\overrightarrow{\text{R}}}_2\right|$.

Answer to Problem 3.23P

The direction of vector $\left|{\overrightarrow{\text{R}}}_1\right|$ is $288°$ counter clockwise with positive $x$-axis and the direction of vector $\left|{\overrightarrow{\text{R}}}_2\right|$ is $26.6°$ counter clockwise with positive $x$-axis

Explanation of Solution

Given info: The value of vector $\left|{\overrightarrow{\text{R}}}_1\right|$ and $\left|{\overrightarrow{\text{R}}}_2\right|$ is $\text{2}\widehat{\text{i}}-6\widehat{\text{j}}$ and $4\widehat{\text{i}}+\text{2}\widehat{\text{j}}$ respectively.

Write the expression for direction of $\left|{\overrightarrow{\text{R}}}_1\right|$.

$\begin{array}{c}\tan {\theta }_{\left|{\overrightarrow{\text{R}}}_1\right|}=\frac{{\left(\left|{\text{R}}_1\right|\right)}_{\text{y}}}{{\left(\left|{\text{R}}_1\right|\right)}_{\text{x}}}\\ ={\tan }^{-1}\left(\frac{{\left(\left|{\text{R}}_1\right|\right)}_{\text{y}}}{{\left(\left|{\text{R}}_1\right|\right)}_{\text{x}}}\right)\end{array}$ (7)

Here,

${\left(\left|{\text{R}}_1\right|\right)}_{\text{x}}$ is the horizontal component of vector $\left|{\overrightarrow{\text{R}}}_1\right|$.

${\left(\left|{\text{R}}_1\right|\right)}_{\text{y}}$ is the vertical component of vector $\left|{\overrightarrow{\text{R}}}_1\right|$.

Substitute $-6$ for ${\left(\left|{\text{R}}_1\right|\right)}_{\text{y}}$ and $2$ for ${\left(\left|{\text{R}}_1\right|\right)}_{\text{x}}$ in equation (7).

$\begin{array}{c}{\theta }_{\left|{\overrightarrow{\text{R}}}_1\right|}={\tan }^{-1}\left(\frac{-6}{2}\right)\\ ={\tan }^{-1}\left(-3.0\right)\\ =-71.56°\end{array}$

The angle made by the vector with the positive axis is,

${\theta }_{\left|{\overrightarrow{\text{R}}}_1\right|}=360°+\left|{\theta }_{\left|{\overrightarrow{\text{R}}}_1\right|}\right|$

Substitute $-71.56°$ for $\left|{\theta }_{\left|{\overrightarrow{\text{R}}}_1\right|}\right|$ in above equation to get the angle made by the vector with the positive axis.

$\begin{array}{c}{\theta }_{\left|{\overrightarrow{\text{R}}}_1\right|}=360°-71.56°\\ =288°\end{array}$

Write the expression for direction of $\left|{\overrightarrow{\text{R}}}_2\right|$.

$\begin{array}{c}\tan {\theta }_{\left|{\overrightarrow{\text{R}}}_2\right|}=\frac{{\left(\left|{\text{R}}_2\right|\right)}_{\text{y}}}{{\left(\left|{\text{R}}_2\right|\right)}_{\text{x}}}\\ ={\tan }^{-1}\left(\frac{{\left(\left|{\text{R}}_2\right|\right)}_{\text{y}}}{{\left(\left|{\text{R}}_2\right|\right)}_{\text{x}}}\right)\end{array}$ (8)

Here,

${\left(\left|{\text{R}}_2\right|\right)}_{\text{x}}$ is the horizontal component of vector $\left|{\overrightarrow{\text{R}}}_2\right|$.

$\left|{\text{R}}_2\right|$ is the vertical component of vector $\left|{\overrightarrow{\text{R}}}_2\right|$.

Substitute $2$ for ${\left(\left|{\text{R}}_2\right|\right)}_{\text{x}}$ and $4$ for $\left|{\text{R}}_2\right|$ in equation (8).

$\begin{array}{c}{\theta }_{\left|{\overrightarrow{\text{R}}}_2\right|}={\tan }^{-1}\left(\frac{2}{4}\right)\\ ={\tan }^{-1}\left(0.5\right)\\ =26.6°\end{array}$

Conclusion:

Therefore, the direction of vector $\left|{\overrightarrow{\text{R}}}_1\right|$ is $288°$ counter clockwise with positive $x$-axis and the direction of vector $\left|{\overrightarrow{\text{R}}}_2\right|$ is $26.6°$ counter clockwise with positive $x$-axis.