Q107 / 180 · COMEDK 2015 · Mathematics

Given, two vectors $\hat{\hat{i}}-\hat{\mathfrak{j}}$ and $\hat{\hat{i}}-2 \hat{\mathfrak{j}}$. The unit vector, coplanar with the two given vectors and perpendicular to $(\hat{\hat{\mathbf{i}}}-\hat{\hat{\mathfrak{j}}})$ is

  • A
    $\frac{1}{\sqrt{2}}(\hat{\mathbf{i}}+\hat{\mathbf{j}})$
  • B
    $\frac{1}{\sqrt{5}}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}})$
  • Correct answer
    C
    $\pm \frac{1}{\sqrt{2}}(\hat{\mathbf{i}}+\hat{\mathbf{j}})$
  • D
    None of these

Solution

Let $=\hat{\mathfrak{i}}-\hat{\mathfrak{j}}$ and $\mathrm{b}=\hat{\mathfrak{i}}-2 \hat{\mathfrak{j}}$ The required vector is along the vector $\mathbf{a} \times(\mathbf{a} \times \mathbf{b})=(\mathbf{a} \cdot \mathbf{b}) \mathbf{a}-(\mathbf{a} \cdot \mathbf{a}) \mathbf{b}$ $$ \begin{aligned} &=3(\hat{\mathbf{i}}-\hat{\mathbf{j}})-2(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}) \\ &=3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}=\hat{\mathbf{i}}+\hat{\mathbf{j}} \end{aligned} $$ Hence, required vectors are given by $\pm \frac{(\hat{\mathbf{i}}+\hat{\mathfrak{j}})}{\sqrt{1+1}}=\pm \cdot \frac{1}{\sqrt{2}}(\hat{\mathbf{i}}+\hat{\mathbf{j}})$

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

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