Vector Algebra
Cross Product Equation — Dot Product Result
nta_pyq_2024_jan
Grade 12
Question:
Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$. If a vector $\vec{p}$ satisfies $\vec{p}\times\vec{b}=\vec{c}\times\vec{b}$ and $\vec{p}\cdot\vec{a}=0$, then $\vec{p}\cdot(\hat{i}-\hat{j}-\hat{k})$ is equal to:
Step-by-Step Solution
Key Concept: $\vec{p}\times\vec{b}=\vec{c}\times\vec{b}\Rightarrow(\vec{p}-\vec{c})\times\vec{b}=\vec{0}\Rightarrow\vec{p}=\vec{c}+\lambda\vec{b}$. Use $\vec{p}\cdot\vec{a}=0$ to find $\lambda$.
$\vec{p}=\vec{c}+\lambda\vec{b}$. $\vec{p}\cdot\vec{a}=0$: $-8+\lambda(-1)=0\Rightarrow\lambda=-8$. $\vec{p}=(1-32,\,-3-8,\,4-56)=(-31,-11,-52)$. $\vec{p}\cdot(\hat{i}-\hat{j}-\hat{k})=-31+11+52=32$.
Correct Answer: 4