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

• A. perpendicular to $b-c$
• B. parallel to $b-c$
• C. parallel to $c+b$
• D. None of these

Answer 1

The correct option is A perpendicular to $b-c$

Consider the problem

$\begin{array}{c}a\times b=c\times d----\left(1\right)\\ \\ a\times c=b\times d----\left(2\right)\end{array}$

Subtracting $\left(2\right)$ from $\left(1\right)$

$\left(a\times b\right)-\left(a\times c\right)=\left(c\times d\right)-\left(b\times d\right)$

$a\times \left(b-c\right)=\left(c-b\right)\times d$

$a\times \left(b-c\right)-\left(c-b\right)\times d=0$

$a\times \left(b-c\right)+\left(b-c\right)\times d=0$

$a\times \left(b-c\right)-d\times \left(b-c\right)=0$

$(a-d)\times (b-c)=0$

Hence, $(a-d)$ is perpendicular to $(b-c)$