Question 77

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

Solution

Given: $$\vec{a} = (1,4,2)$$, $$\vec{b} = (3,-2,7)$$, $$\vec{c} = (2,-1,4)$$. $$\vec{d} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{d} \cdot \vec{a} = 24$$.

$$(\vec{d} - \vec{c}) \times \vec{b} = \vec{0}$$, so $$\vec{d} - \vec{c} = \lambda\vec{b}$$ for some scalar $$\lambda$$.

$$\vec{d} = \vec{c} + \lambda\vec{b} = (2+3\lambda, -1-2\lambda, 4+7\lambda)$$ $$\vec{d} \cdot \vec{a} = (2+3\lambda)(1) + (-1-2\lambda)(4) + (4+7\lambda)(2) = 24$$ $$= 2 + 3\lambda - 4 - 8\lambda + 8 + 14\lambda = 6 + 9\lambda = 24$$

$$9\lambda = 18$$, $$\lambda = 2$$.

$$\vec{d} = (8, -5, 18)$$ $$|\vec{d}|^2 = 64 + 25 + 324 = 413$$

The correct answer is Option D: $$413$$.

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

Solution

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