Vector Algebra
Finding a vector from cross and dot product constraints
nta_pyq_2023_jan
Grade 12
Question:
Let $\vec{a} = \hat{i}+2\hat{j}+3\hat{k}$, $\vec{b} = \hat{i}-\hat{j}+2\hat{k}$ and $\vec{c} = 5\hat{i}-3\hat{j}+3\hat{k}$ be three vectors. If $\vec{r}$ is a vector such that $\vec{r}\times\vec{b} = \vec{c}\times\vec{b}$ and $\vec{r}\cdot\vec{a}=0$. Then $25|\vec{r}|^2$ is equal to
Step-by-Step Solution
Key Concept: From $\vec{r}\times\vec{b} = \vec{c}\times\vec{b}$: $\vec{r} = \vec{c}+\lambda\vec{b}$. Use $\vec{r}\cdot\vec{a}=0$ to find $\lambda$.
$\lambda=-8/5$. $\vec{r} = (17\hat{i}-7\hat{j}-\hat{k})/5$. $25|\vec{r}|^2 = 289+49+1 = 339$. Answer: (3)
Correct Answer: 339