Let a = 3 i ^ + j ^ - k ^ and c = 2 i ^ - 3 j ^ + 3 k ^ . If b is a vector such that a = b × c and | b | 2 = 50 , then | 72 - | b + c | 2 | is equal to ______ . [2023]

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Solution

Q.
Let $\vec{a} = 3 \hat{i} + \hat{j} - \hat{k}$ and $\vec{c} = 2 \hat{i} - 3 \hat{j} + 3 \hat{k} .$ If $\vec{b}$ is a vector such that $\vec{a} = \vec{b} \times \vec{c}$ and $| \vec{b} |^{2}$ $= 50,$ then $| 72 - | \vec{b} + \vec{c} |^{2} |$ is equal to ______ . [2023]

Ans.
(66)

$Given, \vec{a} = 3 \hat{i} + \hat{j} - \hat{k} and \vec{c} = 2 \hat{i} - 3 \hat{j} + 3 \hat{k}$

$| \vec{a} |$ $= \sqrt{9 + 1 + 1} = \sqrt{11} and$ $| \vec{c} |$ $= \sqrt{4 + 9 + 9} = \sqrt{22}$

$Now, \vec{a} = \vec{b} \times \vec{c} \Rightarrow$ $| \vec{a} |$ $=$ $| \vec{b} \times \vec{c} |$ $=$ $| \vec{b} |$ $| \vec{c} |$ $\sin θ$

$\Rightarrow$ $\sqrt{11} = \sqrt{50} \sqrt{22} \sin θ$ $[∵ | \vec{b} | = \sqrt{50}]$

$\Rightarrow \sin θ = \frac{1}{10}$ $and$ $| \vec{b} + \vec{c} |^{2}$ $=$ $| \vec{b} |^{2}$ $+$ $| \vec{c} |^{2}$ $+ 2 \vec{b} \cdot \vec{c}$

$=$ $| \vec{b} |^{2}$ $+$ $| \vec{c} |^{2}$ $+$ $2$ $| \vec{b} |$ $| \vec{c} |$ $\cos θ$

$= 50 + 22 + 2 \times \sqrt{50} \times \sqrt{22} \times \frac{\sqrt{99}}{10}$ $= 72 + 66 = 138$

$∴$ $| 72 - | \vec{b} + \vec{c} |^{2} |$ $=$ $| 72 - 138 |$ $= 66$

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