Vector Algebra
Cross Product Condition — Finding Unknown Vector
nta_pyq_2026_jan
Grade 12
Question:
Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0}$ and $(\vec{a}-\vec{b})\cdot\vec{c}=-97$, then $|\vec{c}\times\hat{k}|^2$ is equal to:
Step-by-Step Solution
Key Concept: $(2\vec{a}+3\vec{b})\times\vec{c}=\vec{0}\Rightarrow\vec{c}\parallel(2\vec{a}+3\vec{b})=(7,-13,19)$. Let $\vec{c}=t(7,-13,19)$.
$|\vec{c}\times\hat{k}|^2=218$.
Correct Answer: 2