Chapter 1, Problem 33P

(a)

To determine

The x and y components of $\text{A}+\text{B}+\text{C}$ .

Answer to Problem 33P

The x components of $\text{A}+\text{B}+\text{C}$ is $7.00$ and y components of $\text{A}+\text{B}+\text{C}$ is $9.00$ .

Explanation of Solution

Given:

Value of $A_x\text{ is }5.00$ , value of $A_y\text{ is }7.00$ ,value of $B_x\text{ is }4.00$ , value of $B_y\text{ is }-3.00$ , value of $C_x\text{ is }-2.00$ and value of $C_y\text{ is }5.00$ .

Formula used:

Vector is the quantity that has both magnitude and direction.

Write the expression for x component of $\text{A}+\text{B}+\text{C}$ .

  $S_x=A_x+B_x+C_x$ …… (1)

Here, $A_x$ is the x component of vector A, $B_x$ is the x component of vector B, $C_x$ is the x component of vector C and $S_x$ is the sum of x components of vector A, vector B and vector C.

Write the expression for y component of $\text{A}+\text{B}+\text{C}$ .

  $S_y=A_y+B_y+C_y$ …… (2)

Here, $A_y$ is the y component of vector A, $B_y$ is the y component of vector B, $C_y$ is the y component of vector C and $S_y$ is the sum of y components of vector A, vector B and vector C.

Calculation:

Substitute $5.00$ for $A_x$ , $4.00$ for $B_x$ and $-2.00$ for $C_x$ in equation (1).

  $\begin{array}{c}{S}_x=5.00+4.00-2.00\\ =7.00\end{array}$

Substitute $7.00$ for $A_y$ , $-3.00$ for $B_y$ and $5.00$ for $C_y$ in equation (2).

  $\begin{array}{c}{S}_y=7.00-3.00+5.00\\ =9.00\end{array}$

Conclusion:

Thus, the x components of $\text{A}+\text{B}+\text{C}$ is $7.00$ and y components of $\text{A}+\text{B}+\text{C}$ is $9.00$ .

(b)

To determine

Magnitude and direction of $\text{A}+\text{B}+\text{C}$ .

Answer to Problem 33P

The magnitude of $\text{A}+\text{B}+\text{C}$ is $11.40$ and the direction of $\text{A}+\text{B}+\text{C}$ is $0.614\widehat{i}+0.789\widehat{j}$ .

Explanation of Solution

Given:

Value of $A_x\text{ is }5.00$ , value of $A_y\text{ is }7.00$ ,value of $B_x\text{ is }4.00$ , value of $B_y\text{ is }-3.00$ , value of $C_x\text{ is }-2.00$ and value of $C_y\text{ is }5.00$ .

Formula used:

Write the expression for vector A.

  $\overrightarrow{A}={\overrightarrow{A}}_x\widehat{i}+{\overrightarrow{A}}_y\widehat{j}$

Write the expression for vector B.

  $\overrightarrow{B}={\overrightarrow{B}}_x\widehat{i}+{\overrightarrow{B}}_y\widehat{j}$

Write the expression for vector C.

  $\overrightarrow{C}={\overrightarrow{C}}_x\widehat{i}+{\overrightarrow{C}}_y\widehat{j}$

Write the expression for sum of vectors.

  $\overrightarrow{S}=\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}$ …… (3)

Substitute ${\overrightarrow{A}}_x\widehat{i}+{\overrightarrow{A}}_y\widehat{j}$ for $\overrightarrow{A}$ , ${\overrightarrow{B}}_x\widehat{i}+{\overrightarrow{B}}_y\widehat{j}$ for $\overrightarrow{B}$ and ${\overrightarrow{C}}_x\widehat{i}+{\overrightarrow{C}}_y\widehat{j}$ for $\overrightarrow{C}$ in equation (3).

  $\overrightarrow{S}=\left({\overrightarrow{A}}_x+{\overrightarrow{B}}_x+{\overrightarrow{C}}_x\right)\widehat{i}+\left({\overrightarrow{A}}_y+{\overrightarrow{B}}_y+{\overrightarrow{C}}_y\right)\widehat{j}$ …… (4)

Write the expression for magnitude of sum.

  $\left|\overrightarrow{S}\right|=\sqrt{{\overrightarrow{S}}^2}$ …… (5)

Here, $\left|\overrightarrow{S}\right|$ is the magnitude of sum of vectors.

Write the expression for direction of sum.

  $\widehat{S}=\frac{\overrightarrow{S}}{\left|\overrightarrow{S}\right|}$ …… (6)

Calculation:

Substitute $5.00$ for $A_x$ , $7.00$ for $A_y$ , $4.00$ for $B_x$ , $-3.00$ for $B_y$ , $-2.00$ for $C_x$ and $5.00$ for $C_y$ in equation (4).

  $\begin{array}{c}\overrightarrow{S}=\left(5.00+4.00-2.00\right)\widehat{i}+\left(7.00-3.00+5.00\right)\widehat{j}\\ =7.00\widehat{i}+9.00\widehat{j}\end{array}$

Substitute $7.00\widehat{i}+9.00\widehat{j}$ for $\overrightarrow{S}$ in equation (5).

  $\begin{array}{c}\left|\overrightarrow{S}\right|=\sqrt{{\left(7.00\right)}^2+{\left(9.00\right)}^2}\\ =11.40\end{array}$

Substitute $7.00\widehat{i}+9.00\widehat{j}$ for $\overrightarrow{S}$ and $11.40$ for $\left|\overrightarrow{S}\right|$ in equation (6).

  $\begin{array}{c}\widehat{S}=\frac{\left(7.00\widehat{i}+9.00\widehat{j}\right)}{11.40}\\ =0.614\widehat{i}+0.789\widehat{j}\end{array}$

Conclusion:

Thus, the magnitude of $\text{A}+\text{B}+\text{C}$ is $11.40$ and the direction of $\text{A}+\text{B}+\text{C}$ is $0.614\widehat{i}+0.789\widehat{j}$ .