To find the magnitude of the resultant vector [tex]\( \mathbf{r} = \mathbf{p} + \mathbf{q} \)[/tex], we need to follow these steps:
1. Decompose each vector into its components:
For vector [tex]\(\mathbf{p}\)[/tex]:
- Magnitude [tex]\( = 7 \)[/tex] units
- Bearing [tex]\( = 52^\circ \)[/tex]
The x-component ([tex]\( p_x \)[/tex]) and y-component ([tex]\( p_y \)[/tex]) can be found using trigonometric functions (cosine and sine, respectively):
[tex]\[
p_x = 7 \cos(52^\circ) = 4.31
\][/tex]
[tex]\[
p_y = 7 \sin(52^\circ) = 5.52
\][/tex]
For vector [tex]\(\mathbf{q}\)[/tex]:
- Magnitude [tex]\( = 12 \)[/tex] units
- Bearing [tex]\( = 163^\circ \)[/tex]
The x-component ([tex]\( q_x \)[/tex]) and y-component ([tex]\( q_y \)[/tex]) can be found similarly:
[tex]\[
q_x = 12 \cos(163^\circ) = -11.48
\][/tex]
[tex]\[
q_y = 12 \sin(163^\circ) = 3.51
\][/tex]
2. Sum the vector components to get the components of the resultant vector [tex]\(\mathbf{r}\)[/tex]:
[tex]\[
r_x = p_x + q_x = 4.31 + (-11.48) = -7.17
\][/tex]
[tex]\[
r_y = p_y + q_y = 5.52 + 3.51 = 9.02
\][/tex]
3. Calculate the magnitude of the resultant vector [tex]\(\mathbf{r}\)[/tex]:
The magnitude [tex]\( |\mathbf{r}| \)[/tex] is given by the Pythagorean theorem:
[tex]\[
|\mathbf{r}| = \sqrt{r_x^2 + r_y^2}
\][/tex]
[tex]\[
|\mathbf{r}| = \sqrt{(-7.17)^2 + (9.02)^2} = \sqrt{51.40 + 81.36} = \sqrt{132.76} \approx 11.5
\][/tex]
Thus, the magnitude of [tex]\(\mathbf{p} + \mathbf{q}\)[/tex] is [tex]\( 11.5 \)[/tex] units, correct to one decimal place.
