Chapter 3, Problem 7P

Figure 3-33 shows two vectors, $\text{<mover accent="true"> <mo> A </mo> <mo> → </mo> </mover>}$ and $\text{<mover accent="true"> <mo> B </mo> <mo> → </mo> </mover>}$ , whose magnitudes are $A=6.8$ units and $B=5.5$ units. Determine $\text{<mover accent="true"> <mo> C </mo> <mo> → </mo> </mover>}$ if (a) $\text{<mover accent="true"> <mo> C </mo> <mo> → </mo> </mover>}=\text{<mover accent="true"> <mo> A </mo> <mo> → </mo> </mover> + <mover accent="true"> <mo> B </mo> <mo> → </mo> </mover>}$ , (b) $\text{<mover accent="true"> <mo> C </mo> <mo> → </mo> </mover>}=\text{<mover accent="true"> <mo> A </mo> <mo> → </mo> </mover>}-\text{<mover accent="true"> <mo> B </mo> <mo> → </mo> </mover>}$ , (c) $\text{<mover accent="true"> <mo> C </mo> <mo> → </mo> </mover>}=\text{<mover accent="true"> <mo> B </mo> <mo> → </mo> </mover>}-\text{<mover accent="true"> <mo> A </mo> <mo> → </mo> </mover>}$ . Give the magnitude and direction for each.

y

Figure 3-33

Problem 7.

To determine

Section 1:

$\overrightarrow{C}=\overrightarrow{A}+\overrightarrow{B}$

Answer to Problem 7P

Solution:

Vector $\overrightarrow{C}$ has a magnitude of 1.3 units and has a direction of 0^° to the East. has a magnitude of 1.3 units and has a direction of 0^° to the east.

Explanation of Solution

Given:

The magnitude $\left|\overrightarrow{A}\right|$ of vector $\overrightarrow{A}$ is 6.8 units and the magnitude $\left|\overrightarrow{B}\right|$ of vector $\overrightarrow{B}$ is 5.5 units.

Formula used:

According to the figure, vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ only have x-component, in effect:

       $\overrightarrow{A}=A_x\widehat{i}=\left|\overrightarrow{A}\right|.\cos \left(0°\right)\widehat{i}=\left|\overrightarrow{A}\right|\widehat{i} \left(1\right)$

This vector points to the right, while that vector $\overrightarrow{B}$points to the left, therefore:

       $\overrightarrow{B}=B_x\widehat{i}=\left|\overrightarrow{B}\right|\cos \left(180˚\right)\widehat{i}=-\left|\overrightarrow{B}\right|\widehat{i} \left(2\right)$

Calculation:

Replacing data given in equations (1)and (2)we obtain:

       $\overrightarrow{A}=6.8 units \widehat{i} \left(3\right)$

and

       $\overrightarrow{B}=-5.5 units \widehat{i} \left(4\right)$

Now, vector $\overrightarrow{C}$ is:

       $\overrightarrow{C}=\overrightarrow{A}+\overrightarrow{B}=6.8 units \widehat{i}+\left(=-5.5 units \widehat{i}\right)=1.3 units \widehat{i} \left(5\right)$

Therefore, vector $\overrightarrow{C}$ has a magnitude of 1.3 units and has a direction of 0^° points to the East.

To determine

Section 2:

$\overrightarrow{C}=\overrightarrow{A}-\overrightarrow{B}$

Answer to Problem 7P

Solution:

Vector $\overrightarrow{C}$ has a magnitude of 12.3 units and has a direction of 0° to the East.

Explanation of Solution

Given:

Results (3)and (4).

Formula used:

The vector $\overrightarrow{C}$ can be obtained using:

       $\overrightarrow{C}=\overrightarrow{A}-\overrightarrow{B} \left(6\right)$

Inserting (3)and (4)into (6)we have:

       $\overrightarrow{C}=6.8 units \widehat{i}-\left(-5.5 units\right)\widehat{i} \left(7\right)$

Solving we obtain:

      $\overrightarrow{C}=12.3 units \widehat{i}$                                                         (8)

Therefore, vector $\overrightarrow{C}$

has a magnitude of 12.3 units and has a direction of 0° to the East.

To determine

Section 3:

$\overrightarrow{C}=\overrightarrow{B}-\overrightarrow{A}$

Answer to Problem 7P

Solution:

Vector $\overrightarrow{C}$ has a magnitude of 12.3 units and has a direction of 180^° to the West.

Explanation of Solution

Explanation.:

Given:

Results (3)and (4).

Formula used:

The vector $\overrightarrow{C}$ can be obtained using:

      $\overrightarrow{C}=\overrightarrow{B}-\overrightarrow{A} \left(9\right)$

Inserting (3)and (4)into (9)we have:

      $\overrightarrow{C}=-5.5 units\widehat{i}-6.8 units \widehat{i} \left(10\right)$

Solving we obtain:

      $\overrightarrow{C}=-12.3 units \widehat{i}\left(11\right)$

Therefore, vector $\overrightarrow{C}$

has a magnitude of 12.3 units and has a direction of 180^° to the West.