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Question

Let $$\vec{a} = 3\hat{i} + 2\hat{j} + x\hat{k}$$ and $$\vec{b} = \hat{i} - \hat{j} + \hat{k}$$, for some real $$X$$. Then $$|\vec{a} \times \vec{b}| = r$$ is possible if:


A
332<r<532
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B
0<r32
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C
32<r332
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D
r532
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Solution

The correct option is C $$r \ge 5 \sqrt{\dfrac{3}{2}}$$
$$\vec{a} \times \vec{b} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k} \\3&2 &x \\1&-1&1\end{vmatrix}$$
$$= (2 +x)\hat{i} + (x - 3)\hat{j} - 5k$$
$$|\vec{a} \times \vec{b}| = \sqrt{4 v+ x^2 + 4x + x^2 + 9 - 6x + 25}$$
$$= \sqrt{2x^2 - 2x + 38}$$
$$\Rightarrow |\vec{a} \times \vec{b}| \ge \sqrt{\dfrac{75}{2}}$$
$$\Rightarrow |\vec{a} \times \vec{b}| \ge 5 \sqrt{\dfrac{3}{2}}$$

Mathematics

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