If a , b , c are three non-zero vectors and n ^ is a unit vector perpendicular to c such that a = b - n ^ , ( 0 ) and b · c = 12 , then | c × ( a × b ) | is equal to [2023]

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Solution

Q.
If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = α \vec{b} - \hat{n}, (α \neq 0)$ and $\vec{b} \cdot \vec{c} = 12,$ then $| \vec{c} \times (\vec{a} \times \vec{b}) |$ is equal to [2023]

1 9

2 15

3 6

4 12

Ans.
(4)

We have, $| \hat{n} |$ $= 1$

Given, $\hat{n} ⊥ \vec{c},$ $\vec{a} = α \vec{b} - \hat{n}$ and $\vec{b} \cdot \vec{c} = 12$

Now, $\vec{a} \times \vec{b} = (α \vec{b} - \hat{n}) \times \vec{b} = - \hat{n} \times \vec{b}$ $∴$ $\vec{c} \times (\vec{a} \times \vec{b}) = \vec{c} \times (- \hat{n} \times \vec{b})$

$= (\vec{c} \cdot \vec{b}) (- \hat{n}) - {\vec{c} \cdot (- \hat{n})} \vec{b}$ $= - 12 \hat{n} + (0) \vec{b} = - 12 \hat{n}$

$∴$ $| \vec{c} \times (\vec{a} \times \vec{b}) |$ $= 12$

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