Vector Algebra Question - CBSE

Question

If $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}, \vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$, find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 15$.

Accepted Answer

$\vec{a} imes \vec{b} = 32\hat{i} - \hat{j} - 14\hat{k} \Rightarrow \vec{d} = \lambda (32\hat{i} - \hat{j} - 14\hat{k})$. [1.5 Marks]
$\vec{c} \cdot \vec{d} = 9\lambda = 15 \Rightarrow \lambda = 5/3 \Rightarrow \vec{d} = \dfrac{160}{3}\hat{i} - \dfrac{5}{3}\hat{j} - \dfrac{70}{3}\hat{k}$. [1.5 Marks]

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🎯 Official CBSE Marking Scheme:
Calculating cross product $\vec{a} imes \vec{b}$: 1.5 Marks
Evaluating scalar multiplier $\lambda = 5/3$ and vector $\vec{d}$: 1.5 Marks

Question:

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