JEE Main & Advanced
Mathematics
Vector Algebra
Question Bank
Critical Thinking
question_answer
The point of intersection of \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\] and \[\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}\], where \[\mathbf{a}=\mathbf{i}+\mathbf{j}\] and \[\mathbf{b}=2\mathbf{i}-\mathbf{k}\] is [Orissa JEE 2004]
A)\[3\mathbf{i}+\mathbf{j}-\mathbf{k}\]
B)\[3\mathbf{i}-\mathbf{k}\]
C)\[3\mathbf{i}+2\mathbf{j}+\mathbf{k}\]
D)None of these
Correct Answer:
A
Solution :
We have \[\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}\]and \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\]
Adding \[\mathbf{r}\times (\mathbf{a}+\mathbf{b})=0\]i.e., r is parallel to \[\mathbf{a}+\mathbf{b}\]
or \[\mathbf{r}=\lambda (\mathbf{i}+\mathbf{j}+2\mathbf{i}-\mathbf{k})\]
\[\mathbf{r}=\lambda (3\mathbf{i}+\mathbf{j}-\mathbf{k})\] for \[\lambda =1\Rightarrow \mathbf{r}=(3\mathbf{i}+\mathbf{j}-\mathbf{k})\].