Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{d} \cdot \vec{a}=24$, then $|\vec{d}|^{2}$ is equal to :

Given that $\vec{d} \times \vec{b} = \vec{c} \times \vec{b}$, we can rewrite this as: <br/><br/>$(\vec{d} - \vec{c}) \times \vec{b} = \vec{0}$ <br/><br/>This implies that the vector $\vec{d} - \vec{c}$ is a scalar multiple of $\vec{b}$: <br/><br/>$\vec{d} = \vec{c} + \lambda \vec{b}$ <br/><br/>Also, we are given that $\vec{d} \cdot \vec{a} = 24$: <br/><br/>$(\vec{c} + \lambda \vec{b}) \cdot \vec{a} = 24$ <br/><br/>Now, we can find the value of $\lambda$ : <br/><br/>$$\lambda = \frac{24 - \vec{a} \cdot \vec{c}}{\vec{b} \cdot \vec{a}} = \frac{24 - 6}{9} = 2$$ <br/><br/>Therefore, we have : <br/><br/>$\vec{d} = \vec{c} + 2(\vec{b}) = 8\hat{i} - 5\hat{j} + 18\hat{k}$ <br/><br/>Now, we can find the squared magnitude of $\vec{d}$ : <br/><br/>$|\vec{d}|^2 = 64 + 25 + 324 = 413$