Vector Algebra
Cross Product Identity — Dot Product from Constraints
nta_pyq_2026_jan
Grade 12
Question:
Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\vec{a}\times\vec{b}=2(\vec{a}\times\vec{c})$. If $|\vec{a}|=1$, $|\vec{b}|=4$, $|\vec{c}|=2$, and the angle between $\vec{b}$ and $\vec{c}$ is $60^\circ$, then $|\vec{a}\cdot\vec{c}|$ is equal to:
Step-by-Step Solution
Key Concept: $\vec{a}\times(\vec{b}-2\vec{c})=\vec{0}\Rightarrow\vec{b}-2\vec{c}=k\vec{a}$. $\vec{b}\cdot\vec{c}=4\cdot2\cdot\cos60^\circ=4$.
$|\vec{a}\cdot\vec{c}|=1$.
Correct Answer: 1