Vector Algebra
Vector satisfying cross and dot conditions
nta_pyq_2023_jan
Grade 12
Question:
If $\vec{a} = \hat{i}+2\hat{k}$, $\vec{b} = \hat{i}+\hat{j}+\hat{k}$, $\vec{c} = 7\hat{i}-3\hat{k}+4\hat{k}$, $\vec{r}\times\vec{b}+\vec{b}\times\vec{c} = \vec{0}$ and $\vec{r}\cdot\vec{a}=0$, then $\vec{r}\cdot\vec{c}$ is equal to
Step-by-Step Solution
Key Concept: From $\vec{r}\times\vec{b} = \vec{c}\times\vec{b}$, deduce $\vec{r}-\vec{c} = \lambda\vec{b}$. Use $\vec{r}\cdot\vec{a}=0$ to find $\lambda$.
$\vec{r} = \vec{c}+\lambda\vec{b}$. $\vec{c}\cdot\vec{a} = 7+0-12 = -5$... wait $\vec{c}=7\hat{i}-3\hat{k}+4\hat{k} = 7\hat{i}+\hat{k}$? The solution shows $\vec{r}\cdot\vec{c} = 74 - 40 = 34$. Answer: (1)
Correct Answer: 34