Vector Algebra
Unit Vector Along Sum — Finding 3λ
nta_pyq_2024_apr
Grade 12
Question:
Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}$ and $\vec{c}=3\hat{i}-\hat{j}+\lambda\hat{k}$ be three vectors. Let $\vec{r}$ be a unit vector along $\vec{b}+\vec{c}$. If $\vec{r}\cdot\vec{a}=3$, then $3\lambda$ is equal to:
Step-by-Step Solution
Key Concept: $\vec{b}+\vec{c}=5\hat{i}+2\hat{j}+(\lambda-5)\hat{k}$. $\vec{r}=k(\vec{b}+\vec{c})$, $|\vec{r}|=1$. $\vec{r}\cdot\vec{a}=3\Rightarrow k(5+4+(3\lambda-15)+3-2+3\lambda)=k(-6+3\lambda)=3$.
$3\lambda=25$.
Correct Answer: 3