Vector Algebra Question 5
Question 5 - 25 January - Shift 1
Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b} \cdot \vec{c}=0$ and $\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}$, then $(\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})$ is equal to
(1) $\frac{3}{4}$
(2) $\frac{1}{2}$
(3) $-\frac{1}{4}$
(4) $\frac{1}{4}$
Show Answer
Answer: (4)
Solution:
Formula: Properties of Scalar product of vectors, Properties of cross product of vectors
$(\overrightarrow{{}a} \cdot \overrightarrow{{}c}) \overrightarrow{{}b}-(\overrightarrow{{}a} \cdot \overrightarrow{{}b}) \overrightarrow{{}c}=\frac{\overrightarrow{{}b}-\overrightarrow{{}c}}{2}$
$\overrightarrow{{}a} \cdot \overrightarrow{{}c}=\frac{1}{2}, \overrightarrow{{}a} \cdot \overrightarrow{{}b}=\frac{1}{2}$
$\therefore \overrightarrow{{}b} \cdot \overrightarrow{{}d}=\frac{1}{2}$
$(\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})=\vec{a} \cdot(\vec{b} \times(\vec{c} \times \vec{d}))$
$=\overrightarrow{{}a} \cdot((\overrightarrow{{}b} \cdot \overrightarrow{{}d}) \overrightarrow{{}c}-(\overrightarrow{{}b} \cdot \overrightarrow{{}c}) \overrightarrow{{}d})$
$=(\overrightarrow{{}a} \cdot \overrightarrow{{}c})(\overrightarrow{{}b} \cdot \overrightarrow{{}d})=\frac{1}{4}$
