Question

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

Answer 1

P(1, 3, 0) and Q(4, 5, 6) are the given points.
\[\therefore \overrightarrow{PQ} = \left( 4 \hat{i} + 5 \hat{j} + 6 \hat{k} \right) - \left( \hat{i} + 3 \hat{j} + 0 \hat{k} \right) = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \]
\[ \Rightarrow \left| \overrightarrow{PQ} \right| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7\]
∴ Unit vector in the direction of \[\overrightarrow{PQ}\] = \[\frac{\overrightarrow{PQ}}{\left| \overrightarrow{PQ} \right|} = \frac{3 \hat{i} + 2 \hat{j} + 6 \hat{k}}{7} = \frac{1}{7}\left( 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \right)\]