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)

$r\times a=b\times a$
$\Rightarrow (r-b)\times a=\vec{0}$
$\Rightarrow r-b$ is paralllel to $a$
$\therefore r-b=\lambda a$
i.e. $r=b+\lambda a(1)$
Similarly, $r\times b=a\times b$ can be written as
$r=a+\mu b(2)$
For point of intersection of the two lines (1) and (2),
we get $b+\lambda a=a+\mu b$
$\Rightarrow \lambda =\mu =1$
Hence, the required point of intersection is given by $r=a+b=i+j+2i-k=3i+j-k$