Vector Algebra
Scalar triple product; vector identities
nta_pyq_2023_jan
Grade 12
Question:
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
Step-by-Step Solution
Key Concept: Expand $\vec{a}\times(\vec{b}\times\vec{c})$ using BAC-CAB rule, use $\vec{b}\cdot\vec{c}=0$ to find $\vec{a}\cdot\vec{c}$ and $\vec{a}\cdot\vec{b}$, then compute the required product.
$\vec{a}\cdot\vec{c}=\vec{a}\cdot\vec{b}=1/2$, $\vec{b}\cdot\vec{d}=1/2$. $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = (\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d}) - (\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c}) = (1/2)(1/2) - 0 = 1/4$. Answer: (4)
Correct Answer: $\frac{1}{4}$