Vector Algebra Question 12
Question 12 - 30 January - Shift 1
If $\overrightarrow{{}ca}$, $\overrightarrow{{}b}$, $\overrightarrow{{}c}$ are three non-zero vectors and $\overrightarrow{{}n}$ is a unit vector perpendicular to $\overrightarrow{{}c}$ such that $\overrightarrow{{}a}=\alpha \overrightarrow{{}b}-\overrightarrow{{}n}$, $(\alpha \neq 0) \quad$ and $\overrightarrow{{}b} \cdot \overrightarrow{{}c}=12, \quad$ then $|\overrightarrow{{}c} \times(\overrightarrow{{}a} \times \overrightarrow{{}b})|$ is equal to :
(1) 15
(2) 9
(3) 12
(4) 6
Show Answer
Answer: (3)
Solution:
Formula: Properties of Scalar product of vectors, Properties of Cross product of vectors
$ \begin{aligned} & \overrightarrow{{}c} \perp \overrightarrow{{}c} \quad \overrightarrow{{}a}=\alpha \overrightarrow{{}b}-\overrightarrow{{}n} \\ & \overrightarrow{{}b} \cdot \overrightarrow{{}c}=12 \\ & \overrightarrow{{}a} \cdot \overrightarrow{{}c}=\alpha(\overrightarrow{{}b} \cdot \overrightarrow{{}c})-\overrightarrow{{}n} \cdot \overrightarrow{{}c} \\ & \overrightarrow{{}a} \cdot \overrightarrow{{}c}=\alpha(\overrightarrow{{}b} \cdot \overrightarrow{{}c}) \\ & |\overrightarrow{{}c} \times(\overrightarrow{{}a} \times \overrightarrow{{}b})|=|(\overrightarrow{{}c} \cdot \overrightarrow{{}b}) \overrightarrow{{}a}-(\overrightarrow{{}c} \cdot \overrightarrow{{}a}) \overrightarrow{{}b}| \\ & =|(\overrightarrow{{}c} \cdot \overrightarrow{{}b}) \overrightarrow{{}a}-\alpha(\overrightarrow{{}b} \cdot \overrightarrow{{}c}) \overrightarrow{{}b}| \\ & =|(\overrightarrow{{}c} \cdot \overrightarrow{{}b})||\overrightarrow{{}a}-\alpha \overrightarrow{{}b}| \\ & =12 \times(|\overrightarrow{{}n}|) \\ & =12 \times 1 \\ & =12 \end{aligned} $
