Question

If $a,b$and $c$ are three vectors such that $a\times b=c$ and $b\times c=a$, then

• A. $a\not\equiv b\not\equiv c$
• B. $a=b=c$
• C. $a\not\equiv b\not\equiv c\not\equiv 1$
• D. $a,b$and $c$ are orthogonal in pairs.

Answer 1

The correct option is D

$a,b$and $c$ are orthogonal in pairs.

Explanation for the correct option:

Step1. Find the relation between $a$ and $b$:

Given $a=bxc$

$=bx(axb)$ ( given)

$=(b.b)a–(b.a)b$

$\Rightarrow a=|b{|}^{2}a–(b.a)b$

Comparing both the sides

$$\begin{gathered} |b{|}^2=1;(b.a)=0 \\ \Rightarrow |b|=1;(a.b)=0 \end{gathered}$$

Therefore, $a$ is perpendicular to $b$

Step2. Find the relation between $b$ and $c$:

Now, $c=axb$

$$\begin{gathered} =(bxc)xb \\ =(b.b)c–(b.c)b \end{gathered}$$

$\Rightarrow c=|b{|}^{2}c–(b.c)b$

On comparing both the sides

$|b{|}^2=1$and$(b.c)=0$

Therefore, $b$is perpendicular to $c$.

Hence, the correct option is D.