Let a ? = – 5 i ^ + j ^ – 3 k ^ , b ? = i ^ + 2 j ^ – 4 k ^ and c ? = ( ( ( a ? × b ? ) × i ^ ) × i ^ ) × i ^ . Then c ? · ( – i ^ + j ^ + k ^ ) is equal to: [2024]

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Solution

Q.
Let $\vec{a} = - 5 \hat{i} + \hat{j} - 3 \hat{k}$ , $\vec{b} = \hat{i} + 2 \hat{j} - 4 \hat{k}$ and $\vec{c} = (((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$ . Then $\vec{c} \cdot (- \hat{i} + \hat{j} + \hat{k})$ is equal to: [2024]

1 –12

2 –15

3 –13

4 –10

Ans.
(1)

We have, $\vec{a} = - 5 \hat{i} + \hat{j} - 3 \hat{k}$ , $\vec{b} = \hat{i} + 2 \hat{j} - 4 \hat{k}$

$∴ (\vec{a} \times \vec{b}) \times \hat{i} = (\vec{a} \cdot \hat{i}) \vec{b} - (\vec{b} \cdot \hat{i}) \vec{a} = - 5 \vec{b} - \vec{a}$

Now, $((- 5 \vec{b} - \vec{a}) \times \hat{i}) \times \hat{i} = (11 \hat{k} - 23 \hat{j}) \times \hat{i} = 11 \hat{j} - 23 \hat{k}$

$∴ \vec{c} \cdot (- \hat{i} + \hat{j} + \hat{k}) = 11 - 23 = - 12$ .

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