Physics · MOTION IN A PLANE

In the given 3D resolution diagram (Fig. 3.9d), a vector \( \mathbf{A} \) makes angles \( \alpha \), \( \beta \), and \( \gamma \) with the x, y, and z axes respectively. If \( \cos\alpha = \frac{1}{2} \) and \( \cos\beta = \frac{1}{\sqrt{2}} \), where \( 0 \le \gamma \le 90^\circ \), determine the value of \( \cos\gamma \), and calculate the z-component \( A_z \) if the magnitude of the vector is \( |\mathbf{A}| = 10\text{ units} \).

1. A
   \( \cos\gamma = \frac{1}{\sqrt{3}} \), \( A_z = \frac{10}{\sqrt{3}}\text{ units} \)
2. B
   \( \cos\gamma = \frac{1}{2} \), \( A_z = 2.5\text{ units} \)
3. C
   \( \cos\gamma = \frac{1}{2} \), \( A_z = 5\text{ units} \)
4. D
   \( \cos\gamma = \frac{\sqrt{3}}{2} \), \( A_z = 5\sqrt{3}\text{ units} \)