Question

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

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

Answer 1

Answer: (C)

a = 0 or b = c

Answer 2

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