Let a = - i + 2 j + 2 k , b = 8 i + 7 j - 3 k and c be a vector such that a × c = b . If c · ( i + j + k ) = 4 , t h e n | a + c | 2 is equal to [2026]

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Solution

Q.
Let $\vec{a} = - \hat{i} + 2 \hat{j} + 2 \hat{k}$ , $\vec{b} = 8 \hat{i} + 7 \hat{j} - 3 \hat{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c} = \vec{b}$ . If $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4, t h e n | \vec{a} + \vec{c} |^{2}$ is equal to [2026]

1 35

2 30

3 27

4 33

Ans.
(3)

$\vec{a} = - \hat{i} + 2 \hat{j} + 2 \hat{k}$

$\vec{b} = 8 \hat{i} + 7 \hat{j} - 3 \hat{k}$

$\vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$

$\vec{a} \times \vec{c} = \vec{b}$ $\Rightarrow (2 c_{3} - 2 c_{2}) \hat{i} + (c_{3} + 2 c_{1}) \hat{j} - (c_{2} + 2 c_{1}) \hat{k} = 8 \hat{i} + 7 \hat{j} - 3 \hat{k}$

$2 c_{3} - 2 c_{2} = 8, c_{3} + 2 c_{1} = 7, c_{2} + 2 c_{1} = 3$

$(c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$

$\Rightarrow c_{1} + c_{2} + c_{3} = 4,$ $c_{1} = 2, c_{2} = - 1, c_{3} = 3$

$| \vec{a} + \vec{c} |^{2} = | \hat{i} + \hat{j} + 5 \hat{k} |^{2} = 27$

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