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Question: If \(\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} \neq 0,\) where **a, b** and **c** ...

If a×b=b×c≠0,\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} \neq 0, where a, b and c are coplanar vectors, then for some scalar k

A

a+c=kb\mathbf{a} + \mathbf{c} = k\mathbf{b}

B

a+b=kc\mathbf{a} + \mathbf{b} = k\mathbf{c}

C

b+c=ka\mathbf{b} + \mathbf{c} = k\mathbf{a}

D

None of these

Answer

a+c=kb\mathbf{a} + \mathbf{c} = k\mathbf{b}

Explanation

Solution

Since a×b=b×c≠0⇒a×b−b×c=0\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} \neq \mathbf{0} \Rightarrow \mathbf{a} \times \mathbf{b} - \mathbf{b} \times \mathbf{c} = \mathbf{0}

⇒a×b+c×b=0⇒(a+c)×b=0\Rightarrow \mathbf{a} \times \mathbf{b} + \mathbf{c} \times \mathbf{b} = \mathbf{0} \Rightarrow (\mathbf{a} + \mathbf{c}) \times \mathbf{b} = \mathbf{0}

⇒a+c\Rightarrow \mathbf{a} + \mathbf{c} is parallel to b⇒a+c=kb.\mathbf{b} \Rightarrow \mathbf{a} + \mathbf{c} = k\mathbf{b}.