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Determine the magnitude of the vector sum \( V = V_1 + V_2 \) and the angle \( \theta_x \) which \( V \) makes with the positive \( x \)-axis. Complete both graphical and algebraic solutions. Assume \( a = 4 \), \( b = 5 \), \( V_1 = 21 \) units, \( V_2 = 23 \) units, and \( \theta = 59^{\circ} \).

Answers:
[tex]\[
\begin{array}{l}
V = 2.55 \, \text{units}
\end{array}
\][/tex]



Answer :

GinnyAnswer
To determine the magnitude of the vector sum \( \mathbf{V} = \mathbf{V}_1 + \mathbf{V}_2 \) and the angle \( \theta_x \) which \( \mathbf{V} \) makes with the positive \( x \)-axis, we'll break down the vectors into their components and then sum them.

Step-by-Step Solution:

1. Determine the components of \( \mathbf{V}_1 \) and \( \mathbf{V}_2 \):
- \( \mathbf{V}_1 \) has magnitude 21 units and is along the positive \( x \)-axis (\( \theta = 0^\circ \)).

[tex]\[ V_{1x} = V_1 \cos(0^\circ) = 21 \cos(0^\circ) = 21.0 \text{ units} \][/tex]
[tex]\[ V_{1y} = V_1 \sin(0^\circ) = 21 \sin(0^\circ) = 0.0 \text{ units} \][/tex]

- \( \mathbf{V}_2 \) has magnitude 23 units and makes an angle \( \theta = 59^\circ \) with the positive \( x \)-axis.

[tex]\[ V_{2x} = V_2 \cos(59^\circ) = 23 \cos(59^\circ) \approx 11.85 \text{ units} \][/tex]
[tex]\[ V_{2y} = V_2 \sin(59^\circ) = 23 \sin(59^\circ) \approx 19.71 \text{ units} \][/tex]

2. Sum the components to get the resultant vector \( \mathbf{V} \):
- Sum the \( x \)-components:

[tex]\[ V_x = V_{1x} + V_{2x} = 21.0 + 11.85 = 32.85 \text{ units} \][/tex]

- Sum the \( y \)-components:

[tex]\[ V_y = V_{1y} + V_{2y} = 0.0 + 19.71 = 19.71 \text{ units} \][/tex]

3. Calculate the magnitude of the resultant vector \( \mathbf{V} \):

[tex]\[ V = \sqrt{V_x^2 + V_y^2} = \sqrt{(32.85)^2 + (19.71)^2} \approx 38.31 \text{ units} \][/tex]

4. Determine the angle \( \theta_x \) which \( \mathbf{V} \) makes with the positive \( x \)-axis:

[tex]\[ \theta_x = \tan^{-1} \left( \frac{V_y}{V_x} \right) = \tan^{-1} \left( \frac{19.71}{32.85} \right) \approx 30.97^\circ \][/tex]

Conclusion:

The magnitude of the vector sum [tex]\( \mathbf{V} \)[/tex] is approximately 38.31 units, and the angle [tex]\( \theta_x \)[/tex] which [tex]\( \mathbf{V} \)[/tex] makes with the positive [tex]\( x \)[/tex]-axis is approximately 30.97 degrees.

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