OnlineCBT
Physics · MOTION IN A PLANE
हिन्दीIn 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 \( |\mathbf{A}| = 10\text{ units} \).
Correct answer · Option 3
\( \cos\gamma = \frac{1}{2} \), \( A_z = 5\text{ units} \)
Explanation
Using direction cosines \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \implies (\frac{1}{2})^2 + (\frac{1}{\sqrt{2}})^2 + \cos^2\gamma = 1 \implies \frac{1}{4} + \frac{1}{2} + \cos^2\gamma = 1 \implies \cos\gamma = \frac{1}{2} \). The z-component is \( A_z = A \cos\gamma = 10 \times \frac{1}{2} = 5\text{ units} \).
