Question

If $a.b=a.c$ and $a\times b=a\times c$, then correct statement is

• A. a || (b - c)
• B. a $\perp$ (b - c)
• C. a = 0 or b = c
• D. None of these

Answer 1

Answer: (C)

$a.b.=a.c\Rightarrow a.(b-c)=0$
$\Rightarrow a=0$ or $b-c=0$ or $a\perp (b-c)$
$\Rightarrow a=0$ or $b=c$ or $a\perp (b-c)\dots$ (1)
Also $a\times b=a\times c\Rightarrow a\times (b-c)=0$
$\Rightarrow a=0$ or $b-c=0$ or $a\Vert (b-c)$
$\Rightarrow a=0$ or $b=c$ or $a\Vert (b-c)\dots$(2)
Observing to $(1)$ and $(2)$ we find that
$a=0$ or $b=c$