Question

The lines $r=a+\lambda (b\times c)$ and $r=b+\mu (c\times a)$ will intersect if

• A. $a\times c=b\times c$
• B. $a.c=b.c$
• C. $b\times a=c\times a$
• D. None of these

Answer 1

The correct option is B $a.c=b.c$

The lines $r=a+\lambda (b\times c)$ and $r=b+\mu (c\times a)$ pass through points $a$ and $b$ respectively and are parallel to vector $b\times c$ and $a\times c$ respectively.

Therefore, they intersect if $a-b,b\times c$ and $c\times a$ are coplanar and so

$(a-b).\{(b\times c)\times (c\times a)\}=0\Rightarrow (a-b).\{\left[bca\right]c-\left[bcc\right]a\}=0\Rightarrow ((a-b).c)\left[bca\right]=0\Rightarrow a.c-b.c=0\Rightarrow a.c=b.c$