If the position vectors of the points $A$ and $B$ are $2 \hat{i}+3 \hat{j}-\hat{k}$ and $\hat{i}-\hat{j}+2…

If the position vectors of the points $A$ and $B$ are $2 \hat{i}+3 \hat{j}-\hat{k}$ and $\hat{i}-\hat{j}+2 \hat{k}$ respectively, then the unit vector along $\overrightarrow{\mathrm{BA}}$ and in the direction of $\overrightarrow{\mathrm{AB}}$ is
  1. $\frac{1}{\sqrt{14}}(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$
  2. $\frac{1}{\sqrt{26}}(-\hat{\mathrm{i}}-4 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$
  3. $\frac{1}{\sqrt{26}}(-3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+\hat{\mathrm{k}})$
  4. $\frac{1}{\sqrt{22}}(3 \hat{i}-4 \hat{j}+3 \hat{k})$

Solution

$\overrightarrow{\mathrm{AB}}=-\hat{i}-4 \hat{j}+3 \hat{k}$ So, $\overrightarrow{\mathrm{BA}}=\hat{i}-4 \hat{j}+3 \hat{k}$ Unit vector along $\overrightarrow{\mathrm{BA}}$ and in direction $\overrightarrow{\mathrm{AB}}$ is $=\frac{\overrightarrow{\mathrm{AB}}}{|\overrightarrow{\mathrm{BA}}|}$ $=\frac{-\hat{i}-4 \hat{j}+3 \hat{k}}{\sqrt{1+16+9}}=\frac{-\hat{i}-4 \hat{j}+3 \hat{k}}{\sqrt{26}}$

Asked in: AP EAMCET 2023 (17 May Shift 2)

Practice more Vectors questions on Aicharya