Question

If \( \bar{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}, \bar{b} = \hat{i} - 2\hat{j} - 2\hat{k}, \bar{c} = -\hat{i} + 4\hat{j} + 3\hat{k} \) and if \( \bar{d} \) is vector perpendicular to both \( \bar{b} \) and \( \bar{c} \), \( \bar{a} \cdot \bar{d} = 18 \), then \( |\bar{a} \times \bar{d}|^2 = \)}

• A. 640
• B. 680
• C. 720
• D. 740

Hint:

Lagrange's Identity: $|\bar{a} \times \bar{b}|^2 + (\bar{a} \cdot \bar{b})^2 = |\bar{a}|^2 |\bar{b}|^2$.

Answer 1

The Correct Option is C

Step 1: Find Direction of \( \vec{d} \)

\( \vec{d} \) is parallel to \( \vec{b} \times \vec{c} \).

\[ \vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & -2 \\ -1 & 4 & 3 \end{vmatrix} = 2\hat{i} - \hat{j} + 2\hat{k} \]

Step 2: Find Magnitude

\[ \vec{d} = \lambda (2\hat{i} - \hat{j} + 2\hat{k}) \]

\[ \vec{a} \cdot \vec{d} = \lambda (4 - 3 + 8) = 18 \;\Rightarrow\; 9\lambda = 18 \;\Rightarrow\; \lambda = 2 \]

\[ \vec{d} = 4\hat{i} - 2\hat{j} + 4\hat{k}, \quad |\vec{d}|^2 = 16 + 4 + 16 = 36 \]

Step 3: Use Lagrange's Identity

\[ |\vec{a} \times \vec{d}|^2 = |\vec{a}|^2 |\vec{d}|^2 - (\vec{a} \cdot \vec{d})^2 \]

\[ |\vec{a}|^2 = 4 + 9 + 16 = 29 \]

\[ = (29)(36) - (18)^2 = 1044 - 324 = 720 \]

Step 4: Conclusion

Hence, the value is \( 720 \).

Final Answer: (C)