Vector Algebra
Cross Product Equation — Finding Scalar Expression
nta_pyq_2024_jan
Grade 12
Question:
Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=3(\hat{i}-\hat{j}+\hat{k})$. Let $\vec{c}$ be a vector such that $\vec{a}\times\vec{c}=\vec{b}$ and $\vec{a}\cdot\vec{c}=3$. Then $\vec{a}\cdot\left[(\vec{c}\times\vec{b})-\vec{b}-\vec{c}\right]$ is equal to:
Step-by-Step Solution
Key Concept: $\vec{a}\cdot[(\vec{c}\times\vec{b})-\vec{b}-\vec{c}]=\vec{a}\cdot(\vec{c}\times\vec{b})-\vec{a}\cdot\vec{b}-\vec{a}\cdot\vec{c}$. Use $\vec{a}\cdot(\vec{c}\times\vec{b})=[\vec{a},\vec{c},\vec{b}]=(\vec{a}\times\vec{c})\cdot\vec{b}=\vec{b}\cdot\vec{b}=|\vec{b}|^2=27$.
$|\vec{b}|^2=9+9+9=27$. $(\vec{a}\times\vec{c})\cdot\vec{b}=\vec{b}\cdot\vec{b}=27$. $\vec{a}\cdot\vec{b}=3-6+3=0$. $\vec{a}\cdot\vec{c}=3$. Answer $=27-0-3=24$.
Correct Answer: 2