Q75 / 125 · BITSAT 2010 · Mathematics

With respect to a rectangular cartesian coordinate system, three vectors are expressed as : \(\vec{a}=4 \hat{i}-\hat{j}, \vec{b}=-3 \hat{i}+2 \hat{j}\) and \(\vec{c}=-\hat{k} \quad\) where \(\hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}\) are unit vectors, along the \(X\), \(Y\) and \(Z\)-axis respectively. The unit vector \(\hat{r}\) along the direction of sum of these vector is -

  • Correct answer
    A
    \(\hat{\mathrm{r}}=\frac{1}{\sqrt{3}}(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})\)
  • B
    \(\hat{\mathrm{r}}=\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})\)
  • C
    \(\hat{\mathrm{i}}=\frac{1}{3}(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})\)
  • D
    \(\hat{\mathrm{r}}=\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)

Solution

\(\begin{aligned} & \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=4 \hat{\mathrm{i}}-\hat{\mathrm{j}}-3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}} \\ & \hat{\mathrm{r}}=\frac{\overrightarrow{\mathrm{r}}}{|\mathrm{r}|}=\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{1^2+1^2+(-1)^2}}=\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{3}} \end{aligned}\)

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

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