Q97 / 200 · Manipal (MET) 2010 · Mathematics

Let $\overrightarrow{\mathbf{u}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \overrightarrow{\mathbf{v}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}$ and $\overrightarrow{\mathbf{w}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$, if $\hat{\mathbf{n}}$ is a unit vector such that $\overrightarrow{\mathbf{u}} \cdot \hat{\mathbf{n}}=0$ and $\overrightarrow{\mathbf{v}} \cdot \hat{\mathbf{n}}=0$, then $|\overrightarrow{\mathbf{w}} \cdot \hat{\mathbf{n}}|$ is equal to

  • Correct answer
    A
    3
  • B
    0
  • C
    1
  • D
    2

Solution

We have, $\overrightarrow{\mathbf{u}} \cdot \hat{\mathbf{n}}=0$ and $\overrightarrow{\mathbf{v}} \cdot \hat{\mathbf{n}}=0$ $\Rightarrow \hat{\mathbf{n}} \perp \overrightarrow{\mathbf{u}}$ and $\hat{\mathbf{n}} \perp \overrightarrow{\mathbf{v}}$ $\Rightarrow \hat{\mathbf{n}}= \pm \frac{\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}}{|\hat{\mathbf{u}} \times \overrightarrow{\mathbf{v}}|}$ Now, $\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}) \times(\hat{\mathbf{i}}-\hat{\mathbf{j}})=-2 \hat{\mathbf{k}}$ $\hat{\mathbf{n}}= \pm \hat{\mathbf{k}}$ Hence, $|\overrightarrow{\mathbf{w}} \cdot \hat{\mathbf{n}}|=|(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \cdot( \pm \hat{\mathbf{k}})|=3$

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

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