Trigonometry & Inverse Trigonometry
Vector Triangle — Dot Product of Sides
nta_pyq_2023_apr
Grade 11
Question:
Let $\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}$, $\overrightarrow{CB}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$, $\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta\hat{k}$. If $\delta>0$ and area of $\triangle ABC=5\sqrt{6}$, then $\overrightarrow{CB}\cdot\overrightarrow{CA}$ is equal to
Step-by-Step Solution
Key Concept: From $\overrightarrow{CA}+\overrightarrow{AB}=\overrightarrow{CB}$: $\alpha=2,\ \beta=4,\ \gamma=\delta+3$. Area $=\frac{1}{2}|\overrightarrow{AB}\times\overrightarrow{BC}|=5\sqrt{6}$.
$\gamma=8,\ \delta=5$. $\overrightarrow{CB}\cdot\overrightarrow{CA}=60$.
Correct Answer: 1