Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$, and $\vec{a} \cdot \vec{d}=18$, then $|\vec{a} \times \vec{d}|^{2}$ is equal to :

<p>Given vectors : <br/><br/>$ \vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} $ <br/><br/>$ \vec{b} = \hat{i} - 2\hat{j} - 2\hat{k} $ <br/><br/>$ \vec{c} = -\hat{i} + 4\hat{j} + 3\hat{k} $</p> <p>Since $ \vec{d} $ is perpendicular to both $ \vec{b} $ and $ \vec{c} $, its direction is given by their cross product :</p> <p>$ \vec{d} = \lambda(\vec{b} \times \vec{c}) $</p> <p>$$ \begin{aligned} \vec{b} \times \vec{c} & =\left|\begin{array}{ccc} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 1 & -2 & -2 \\ -1 & 4 & 3 \end{array}\right| \\\\ & =\hat{\mathbf{i}}(-6+8)-\hat{\mathbf{j}}(3-2)+\hat{\mathbf{k}}(4-2)=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}} \end{aligned} $$</p> <p>Thus, $ \vec{b} \times \vec{c} = 2\hat{i} - \hat{j} + 2\hat{k} $</p> <p>Given this, $ \vec{d} $ can be expressed as : <br/><br/>$ \vec{d} = \lambda(2\hat{i} - \hat{j} + 2\hat{k}) $</p> <p>Using the given condition that $ \vec{a} \cdot \vec{d} = 18 $ : <br/><br/>$ (2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot \lambda(2\hat{i} - \hat{j} + 2\hat{k}) = 18 $</p> <p>$ \Rightarrow 2\lambda(2) - 3\lambda(1) + 4\lambda(2) = 18 $ <br/><br/>$ \Rightarrow \lambda(4 + 8 - 3) = 18 $ <br/><br/>$ \Rightarrow 9\lambda = 18 $ <br/><br/>$ \Rightarrow \lambda = 2 $</p> <p>So, $ \vec{d} = 4\hat{i} - 2\hat{j} + 4\hat{k} $</p> <p>Now, using the identity : <br/><br/>$ |\vec{a} \times \vec{d}|^2 = |\vec{a}|^2 |\vec{d}|^2 - (\vec{a} \cdot \vec{d})^2 $</p> <p>Given $ |\vec{a}| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{29} $ <br/><br/>and $ |\vec{d}| = \sqrt{4^2 + (-2)^2 + 4^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6$</p> <p>Using your corrected calculations : <br/><br/>$ |\vec{a} \times \vec{d}|^2 = 29 \times 36 - 18^2 = 1044 - 324 = 720 $</p>