Q23 / 66 · JEE Advanced 2007 (Paper 2) · Mathematics

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be unit vectors such that $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$. Which one of the following is correct?

  • A
    $\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{c} \times \mathbf{a}=\mathbf{0}$
  • Correct answer
    B
    $\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{c} \times \mathbf{a} \neq \mathbf{0}$
  • C
    $\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{a} \times \mathbf{c}=\mathbf{0}$
  • D
    $\mathbf{a} \times \mathbf{b}, \mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}$ are mutually perpendicular

Solution

Since $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors and $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$ $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ represent an equilateral triangle. $$ \therefore \quad \mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{c} \times \mathbf{a} \neq \mathbf{0} . $$

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

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