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Question

Find the magnitude of projection of vector $$2i + 3j + k$$, on a vector which is perpendicular to the plane containing vectors $$i + j + k$$ and $$i + 2j + 3k$$.


A
32
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B
23
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C
43
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D
223
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Solution

The correct option is A $$\dfrac{\sqrt{3}}{\sqrt{2}}$$
Normal vector to the plane to plane containing $$\hat{i} + \hat{j} + \hat{k}$$ and $$\hat{i} + 2 \hat{j} + 3 \hat{k}$$ is 
$$\bar{n} = (\hat{i} + \hat{j} + \hat{k}) \times (\hat{i} + 2\hat{j} + 3 \hat{k})$$
$$\bar{n} = \hat{i} - 2 \hat{j} + \hat{k}$$
projection of $$(2\hat{i} + 3 \hat{j} + \hat{k})$$ on $$\bar{n}$$
$$= \left|\dfrac{(2 \hat{i} + 3 \hat{j} + \hat{k}) . (\hat{i} - 2 \hat{j} + \hat{k})}{\sqrt{1 + 4 + 1}}\right|$$
$$= \dfrac{3}{\sqrt{6}} = \sqrt{\dfrac{3}{2}}$$

Physics

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