Question

Write two different vectors having same direction.

Answer 1

Let \[\overrightarrow{p} = \hat{i} + 2\hat{j} + 3 \hat{k}\] and \[\overrightarrow{q} = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}\]
Then, direction cosines of \[\vec{p}\] are
\[l = \frac{1}{\sqrt{1^2 + 2^2 + 3^2}} = \frac{2}{\sqrt{14}} , m = \frac{2}{\sqrt{1^2 + 2^2 + 3^2}} = \frac{2}{\sqrt{14}}\text{ and }n = \frac{3}{\sqrt{1^2 + 2^2 + 3^2}} = \frac{3}{\sqrt{14}}\]
Direction cosines of \[\overrightarrow{q}\] are\[l = \frac{2}{\sqrt{2^2 + 4^2 + 6^2}} = \frac{2}{2\sqrt{14}} = \frac{1}{\sqrt{14}} , m = \frac{4}{\sqrt{2^2 + 4^2 + 6^2}} = \frac{4}{2\sqrt{14}} = \frac{2}{\sqrt{14}} \text{ and }n = \frac{6}{\sqrt{2^2 + 4^2 + 6^2}} = \frac{6}{2\sqrt{14}} = \frac{3}{\sqrt{14}}\]
The direction cosines of two vectors are same. Hence the two diffrent vectors \[\overrightarrow{p} , \overrightarrow{q}\] have same directions.