Question

If $a\times b=c\times d$ and $a\times c=b\times d$, then $a-d$ $(a\ne d$ and $b\ne c)$ is, parallel to

• A. $b-c$
• B. $c+a$
• C. $c+b$
• D. None of these

Answer 1

Answer: (A)

$(a-d)\times (b-c)=0$
$\Rightarrow a\times b-a\times c-d\times b+d\times c=0$
Using $a\times b=c\times d$
and $a\times c=b\times d$
we get, $c\times d-b\times d-d\times b+d\times c=0$
$\Rightarrow d\times c+d\times b-d\times b+d\times c=0$
(as $a\times b=-b\times a)$
$\Rightarrow 0=0$
$\therefore (a-d)\times (b-c)=0$
which shows $(a-d)$ is parallel to $(b-c)$, when $a\ne b$ and
$b\ne c$