Vector Algebra
Cross Product — Projection of One Vector on Another
nta_pyq_2026_jan
Grade 12
Question:
Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$, $\vec{c}=\lambda\hat{i}+\hat{j}+\hat{k}$ and $\vec{v}=\vec{a}\times\vec{b}$. If $\vec{v}\cdot\vec{c}=11$ and the length of the projection of $\vec{b}$ on $\vec{c}$ is $p$, then $9p^2$ is equal to:
Step-by-Step Solution
Key Concept: $\vec{v}=\vec{a}\times\vec{b}=-\hat{i}+7\hat{j}+5\hat{k}$. $\vec{v}\cdot\vec{c}=-\lambda+7+5=11\Rightarrow\lambda=1$. $\vec{c}=\hat{i}+\hat{j}+\hat{k}$, $|\vec{c}|=\sqrt{3}$.
$9p^2=12$.
Correct Answer: 1