Q77 / 170 · Manipal (MET) 2017 · Mathematics

$\mathbf{a}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}), \mathbf{b}=\frac{1}{7}(3 \hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}+2 \hat{\mathbf{k}}) . \text { If } \mathbf{a}$ and $\mathbf{b}$ are mutually perpendicular, then value of $\lambda$ is

  • A
    2
  • B
    -1
  • Correct answer
    C
    6
  • D
    -6

Solution

Given, $\mathbf{a}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}), \mathbf{b}=\frac{1}{7}(3 \hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ Since, the vectors $\mathbf{a}$ and $\mathbf{b}$ are perpendicular. $\begin{array}{ll} \therefore & \mathbf{a} \cdot \mathbf{b}=0 \\ \Rightarrow & \frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}) \cdot \frac{1}{7}(3 \hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}+2 \hat{\mathbf{k}})=0 \\ \Rightarrow & \frac{1}{49}(6-3 \lambda+12)=0 \\ \Rightarrow & 3 \lambda=18 \\ \Rightarrow & \lambda=6 \end{array}$

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

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