Question

Let $a=i+j$ and $b=2i-k,$ the point of intersection of the lines $r\times a=b\times a$ and $r\times b=a\times b$ is

• A. $-i+j+k$
• B. $3i-j+k$
• C. $3i+j-k$
• D. $i-j-k$

Answer 1

Answer: (C)

$3i+j-k$

$\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}\Rightarrow \overrightarrow{r}=\overrightarrow{b}+\lambda \overrightarrow{a}$
and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}\Rightarrow \overrightarrow{r}=\overrightarrow{a}+\mu \overrightarrow{b}$
For intersection of both the lines, $\overrightarrow{b}+\lambda \overrightarrow{a}=\overrightarrow{a}+\mu \overrightarrow{b}$
Comparing coefficients, $\lambda =\mu =1$
Hence point of intersection is $\overrightarrow{r}=\overrightarrow{a}+\overrightarrow{b}=3\hat{i}+\hat{j}-\hat{k}$