Q86 / 200 · Manipal (MET) 2013 · Mathematics

If $\mathbf{a} \times \mathbf{b}=\mathbf{c}, \mathbf{b} \times \mathbf{c}=\mathbf{a}$ and $a, b, c$ be the moduli of the vectors $\mathbf{a}, \mathbf{b}$, c respectively, then

  • A
    $a=1, b=1$
  • B
    $c=1, a=1$
  • C
    $a \cdot(b \times c)=1$
  • Correct answer
    D
    $b=1, c=a$

Solution

Since, $\mathbf{a} \times \mathbf{b}=\mathbf{c}$ $\begin{aligned} & \therefore(\mathbf{b} \times \mathbf{c}) \times \mathbf{b}=\mathbf{c} \\ & \Rightarrow(\mathbf{b} \cdot \mathbf{b}) \mathbf{c}-(\mathbf{b} \cdot \mathbf{c}) \mathbf{b}=\mathbf{c} \\ & \Rightarrow(\mathbf{b} \cdot \mathbf{b})=1, \quad \mathbf{c} \cdot \mathbf{b}=0 \\ & \Rightarrow \quad b^2=1, \text { i.e., } b=1\end{aligned}$ $\therefore \quad$ b is a unit vector $\therefore|\mathbf{c}|=|\mathbf{a}| \quad \Rightarrow c=a$.

Hence, the correct answer is option: 'D'.

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