Q248 / 270 · NDA 2022 (Phase 1) · Mathematics

Consider the following statements : 1. Dot product over vector addition is distributive 2. Cross product over vector addition is distributive 3. Cross product of vectors is associative Which of the above statements is/are correct?

  • A
    1 only
  • B
    2 only
  • Correct answer
    C
    1 and 2 only
  • D
    1, 2 and 3

Solution

$\because \vec{a} \cdot(\vec{b}+\vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}$ Thus, the dot- product is distributive over addition and $\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$ Thus the cross product is distributive over addition $\begin{aligned} & \because \vec{a} \times(\vec{b}+\vec{c})=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c} \\ & \text {and }(\vec{a} \times \vec{b}) \times \vec{c}=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{c} \cdot \vec{b}) \vec{a} \\ & \therefore \vec{a} \times(b \times \vec{c}) \neq(\vec{a} \times \vec{b}) \times \vec{c}\end{aligned}$ Thus cross product of vectors is not associative

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

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