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Let \(\vec{a}, \vec{b}, \vec{c}\) be any three vectors and m, n be scalars. Which one of the following is not true?

  • A.\((\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})\)
  • B.\(m(\vec{a} + \vec{b} + \vec{c}) = m\vec{a} + m\vec{b} + m\vec{c}\)
  • C.\((m + n)\vec{a} = m\vec{a} + n\vec{a}\)
  • D.\(m(\vec{a}.\vec{b} = m\vec{a}.m\vec{b})\)
  • E.\(m(\vec{a} \times \vec{b} = m\vec{a} \times \vec{b})\)

Answer: D

The statement that is not true is \(m(\vec{a}.\vec{b}) = m\vec{a}.m\vec{b}\).

A scalar can be taken out of a dot product only once: \((m\vec{a}).(m\vec{b}) = m^{2}(\vec{a}.\vec{b})\). So the right side is m times too large unless m is 0 or 1.

The other statements are standard laws: vector addition is associative; scalar multiplication distributes over a sum of vectors and over a sum of scalars; and \(m(\vec{a} \times \vec{b}) = (m\vec{a}) \times \vec{b}\), because the scalar is applied to one factor of the cross product, not both.

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