Question

Write a unit vector in the direction of $\overrightarrow{\mathrm{PQ}}$, where P and Q are the points (1, 3, 0) and (4, 5, 6) respectively. [CBSE 2014]

Answer 1

P(1, 3, 0) and Q(4, 5, 6) are the given points.

$$\begin{gathered} \therefore \overrightarrow{\mathrm{PQ}}=\left(4\hat{i}+5\hat{j}+6\hat{k}\right)-\left(\hat{i}+3\hat{j}+0\hat{k}\right)=3\hat{i}+2\hat{j}+6\hat{k} \\ \Rightarrow \left|\overrightarrow{\mathrm{PQ}}\right|=\sqrt{3^2+2^2+6^2}=\sqrt{9+4+36}=\sqrt{49}=7 \end{gathered}$$

∴ Unit vector in the direction of $\overrightarrow{\mathrm{PQ}}$ = $\frac{\overrightarrow{\mathrm{PQ}}}{\left|\overrightarrow{\mathrm{PQ}}\right|}=\frac{3\hat{i}+2\hat{j}+6\hat{k}}{7}=\frac{1}{7}\left(3\hat{i}+2\hat{j}+6\hat{k}\right)$