Vector Algebra
Scalar Triple Product — Unit Vector Condition
nta_pyq_2024_apr
Grade 12
Question:
Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\vec{c}=x\hat{i}+2\hat{j}+3\hat{k}$, $x\in\mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a}\cdot\vec{d}=1$, then $(\vec{a}\times\vec{b})\cdot\vec{c}$ is equal to
Step-by-Step Solution
Key Concept: $\vec{b}+\vec{c}=(x+2)\hat{i}+6\hat{j}-2\hat{k}$. $\vec{d}=\lambda(\vec{b}+\vec{c})$ (unit vector). $\vec{a}\cdot\vec{d}=1\Rightarrow\lambda(x+2+6-2)=1\Rightarrow\lambda(x+6)=1$ ...(1). $|\vec{d}|=1\Rightarrow\lambda^2((x+2)^2+36+4)=1$ ...(2).
$x=1$. $(\vec{a}\times\vec{b})\cdot\vec{c}=11$.
Correct Answer: 1