Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Let $\vec{a} = \vec{i} + \vec{j}$ and $\vec{b} = 2\vec{i} - \vec{k}$. The point of intersection of the lines $\vec{r} \times \vec{a} = \vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b} = \vec{a} \times \vec{b}$ is:
\vec{i} + \vec{j} - 3\vec{k}
3\vec{i} + \vec{j} + \vec{k}
3\vec{i} + \vec{j} - \vec{k}
None of these
Step-by-Step Solution
Key Concept: Cross product equations determine parallel directions; intersection of two parametric lines solves for specific position.
Given $\vec{r} \times \vec{a} = \vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b} = \vec{a} \times \vec{b}$, we have $(\vec{r} - \vec{b}) \times \vec{a} = 0$ and $(\vec{r} - \vec{a}) \times \vec{b} = 0$. These imply $\vec{r} = \vec{b} + p\vec{a}$ and $\vec{r} = \vec{a} + s\vec{b}$. Equating gives $\vec{b} + p\vec{a} = \vec{a} + s\vec{b}$, which with linear independence yields $p = s = 1$, so $\vec{r} = 3\vec{i} + \vec{j} - \vec{k}$.
Correct Answer: 3