Vector Algebra
Two Statements — Vector $\vec{r}$ and Triangle Inequality
nta_pyq_2024_apr
Grade 12
Question:
Between the following two statements:
Statement I: Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a}\times\vec{r}=\vec{a}\times\vec{b}$ and $\vec{a}\cdot\vec{r}=0$ is of magnitude $\sqrt{10}$.
Statement II: In a triangle $ABC$, $\cos2A+\cos2B+\cos2C\geq-\dfrac{3}{2}$.
Statement I is incorrect but Statement II is correct.
Both Statement I and Statement II are correct.
Statement I is correct but Statement II is incorrect.
Both Statement I and Statement II are incorrect.
Step-by-Step Solution
Key Concept: Statement I: $\vec{a}\times(\vec{r}-\vec{b})=0\Rightarrow\vec{r}-\vec{b}=\lambda\vec{a}\Rightarrow\vec{r}=\vec{b}+\lambda\vec{a}$. $\vec{a}\cdot\vec{r}=0\Rightarrow\vec{a}\cdot\vec{b}+\lambda|\vec{a}|^2=0\Rightarrow7-7\lambda=0\Rightarrow\lambda=-2$... let me check: $\vec{a}\cdot\vec{b}=2+2+3=7$, $|\vec{a}|^2=14$. $\lambda=-1/2$. $\vec{r}=(3\hat{i}+\hat{k})/2$. $|\vec{r}|=\sqrt{10}/2\neq\sqrt{10}$.
Statement I incorrect, Statement II correct.
Correct Answer: 1