Question

If $(a+b),(b+c)$and $(c+a)$ are coplanar then $a\times (b\times c)$ is?

• A. $0$
• B. $1$
• C. $a.b$
• D. $b.c$

Answer 1

The correct option is A

$0$

Find the value of $a\times (b\times c)$:

Given that $(a+b),(b+c)$and $(c+a)$ are coplanar.

Therefore,

$$\begin{gathered} \Rightarrow (a+b)\left[\right(b+c)x(c+a\left)\right]=0 \\ \Rightarrow (a+b)(bxc+bxa+cxc+cxa)=0 \\ \Rightarrow (a+b)(bxc+bxa+cxa)=0 \\ \Rightarrow a.(bxc)+a.(bxa)+a.(cxa)+b.(bxc)+b.(bxa)+b.(cxa)=0 \\ \Rightarrow [abc]+0+0+0+0=0 \\ \Rightarrow [abc]=0 \\ \Rightarrow a\times (b\times c)=0 \end{gathered}$$

Hence, the correct option is (A).