Step 1: Understanding the Question:
We need to find vector \( \bar{d} \) first using the perpendicularity and dot product conditions, then calculate the magnitude squared of the cross product.
Step 2: Key Formula or Approach:
1. \( \bar{d} = \lambda (\bar{b} \times \bar{c}) \).
2. \( \bar{a} \cdot \bar{d} = 18 \implies \lambda \bar{a} \cdot (\bar{b} \times \bar{c}) = 18 \).
3. \( |\bar{a} \times \bar{d}|^2 = |\bar{a}|^2 |\bar{d}|^2 - (\bar{a} \cdot \bar{d})^2 \).
Step 3: Detailed Explanation:
Calculate \( \bar{b} \times \bar{c} \):
\[ \bar{b} \times \bar{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
1 & -2 & -2
-1 & 4 & 3 \end{vmatrix} = \hat{i}(-6 + 8) - \hat{j}(3 - 2) + \hat{k}(4 - 2) = 2\hat{i} - \hat{j} + 2\hat{k} \]
Let \( \bar{d} = \lambda (2, -1, 2) \).
Given \( \bar{a} \cdot \bar{d} = 18 \implies (2, 3, 4) \cdot \lambda(2, -1, 2) = 18 \implies \lambda(4 - 3 + 8) = 18 \implies 9\lambda = 18 \implies \lambda = 2 \).
So, \( \bar{d} = (4, -2, 4) \).
Now, calculate magnitudes:
\( |\bar{a}|^2 = 2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29 \).
\( |\bar{d}|^2 = 4^2 + (-2)^2 + 4^2 = 16 + 4 + 16 = 36 \).
Calculate \( |\bar{a} \times \bar{d}|^2 \):
\[ |\bar{a} \times \bar{d}|^2 = (29)(36) - (18)^2 = 1044 - 324 = 720 \]
Step 4: Final Answer:
The value is 720.