Q108 / 180 · COMEDK 2017 · Mathematics

If $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{b}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$, then find $(a+3 b) \cdot(2 a-b) .$

  • Correct answer
    A
    $-15$
  • B
    12
  • C
    13
  • D
    $-10$

Solution

Here, $\quad \mathbf{a}+3 \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}+3(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})$ $=10 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $2 \mathbf{a}-\mathbf{b}=2(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})-(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})=-\hat{\mathbf{i}}+5 \hat{\mathbf{k}}$ $(\mathbf{a}+3 \mathbf{b}) \cdot(2 \mathbf{a}-\mathbf{b})=(10 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \cdot(-\hat{\mathbf{i}}+5 \hat{\mathbf{k}})$ $=10 \times(-1)+7 \times 0+-1 \times 5=-15$

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

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