Question

Let $a,b$ and $c$ represent vector quantities. Which of the following points are collinear>|?

• A. $a-2b+3c$, $2a+3b-4c$, $-7b+10c$
• B. $-2a+3b+5c$, $a+2b+3c$, $7a-c$
• C. $a$, $b$ , $3a-2b$
• D. $a+b-c$, $3a-4c$, $2a+b+3c$

Answer 1

Answer: (A), (B)

The correct options are
A $a-2b+3c$, $2a+3b-4c$, $-7b+10c$
B $-2a+3b+5c$, $a+2b+3c$, $7a-c$

(a) $\Delta =\left|\begin{array}{ccc}1& -2& 3\\ 2& 3& -4\\ 0& -7& 10\end{array}\right|$
$=7(-4-6)+10(3+4)$
$=-70+70=0$, collinear
(b) $\Delta =\left|\begin{array}{ccc}-2& 3& 5\\ 1& 2& 3\\ 7& 0& -1\end{array}\right|$
$=7(9-10)-1(-4-3)$
$=-7+7=0$, collinear
(c) $\Delta \ne 0$
(d) $\Delta \ne 0$