The vector $(\hat{i} \times \vec{a} \cdot \vec{b}) \hat{i}+(\hat{j} \times \vec{a} \vec{b}) \hat{j}+(\hat{k}…

The vector $(\hat{i} \times \vec{a} \cdot \vec{b}) \hat{i}+(\hat{j} \times \vec{a} \vec{b}) \hat{j}+(\hat{k} \times \vec{a} \cdot \vec{b}) \hat{k}$ is equal to:
  1. $\vec{b} \times \vec{a}$
  2. $\vec{a}$
  3. $\vec{a} \times \vec{b}$
  4. $\vec{b}$

Solution

$(\hat{i} \times \vec{a} \cdot \vec{b}) \hat{i}+(\hat{j} \times \vec{a} \cdot \vec{b}) \hat{j}+(\hat{k} \times \vec{a} \cdot \vec{b}) \hat{k}$ $=(\hat{i} \cdot \vec{a} \times \vec{b}) \hat{i}+(\hat{j} \cdot \vec{a} \times \vec{b}) \hat{j}+(\hat{k} \cdot \vec{a} \times \vec{b}) \hat{k}$ $(\because \vec{a} \times \vec{b} \cdot \vec{c}=\vec{a} \cdot \vec{b} \times \vec{c})$ $=(\vec{a} \times \vec{b}) \hat{i}+(\vec{a} \times \vec{b}) \hat{j}+(\vec{a} \times \vec{b}) \hat{k}$ $==\vec{a} \times \vec{b}$

Asked in: JEE Main 2013 (09 Apr Online)

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