Question

If \( \theta \) is the angle between \( \vec{f} = i + 2j - 3k \) and \( \vec{g} = 2i - 3j + ak \) and \( \sin \theta = \frac{\sqrt{24}}{28} \), then \( 7a^2 + 24a = \) ?

• A. \( 10 \)
• B. \( 12 \)
• C. \( 36 \)
• D. \( 15 \)

Hint:

Use the determinant method to compute the cross product efficiently.

Answer 1

The Correct Option is A

Step 1: Formula for angle between two vectors The formula for the sine of the angle between two vectors is given by: \[ \sin \theta = \frac{|\vec{f} \times \vec{g}|}{|\vec{f}||\vec{g}|} \] 
Step 2: Compute cross product magnitude \( |\vec{f} \times \vec{g}| \) Using determinant method, 

\[\vec{f} \times \vec{g} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -3 \\ 2 & -3 & a \end{vmatrix}\]

 Expanding, \[ \vec{f} \times \vec{g} = \hat{i} (2a + 9) - \hat{j} (a + 6) + \hat{k} (-3 - 4) \] \[ = (2a + 9) \hat{i} - (a + 6) \hat{j} - 7\hat{k} \] \[ |\vec{f} \times \vec{g}| = \sqrt{(2a+9)^2 + (a+6)^2 + 49} \] 
Step 3: Solve for \( a \) using \( \sin \theta \) equation Given \( \sin \theta = \frac{\sqrt{24}}{28} \), solving for \( a \): \[ 7a^2 + 24a = 10 \] Thus, the correct answer is option (1).