Question

Write a vector of magnitude 12 units which makes 45° angle with X-axis, 60° angle with Y-axis and an obtuse angle with Z-axis.

Answer 1

Suppose a vector \[\overrightarrow{r}\] makes an angle \[45^{\circ}\] with OX, \[60^{\circ}\] with \[O\Upsilon\] and having magnitude 12 units.
\[l = \cos 45^{\circ} = \frac{1}{\sqrt{2}}\text{ and }m = \cos 60^{\circ} = \frac{1}{2}\]
\[\text{ Now, }l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow \frac{1}{2} + \frac{1}{4} + n^2 = 1\]
\[ \Rightarrow n^2 = \frac{1}{4}\]
\[ \Rightarrow n = - \frac{1}{2} \left[ \because \text{ The angle with the z - axis is obtuse }\right]\]
Therefore, 
\[\overrightarrow{r} = \left| \overrightarrow{r} \right| \left( l \hat{i} + m \hat{j} + n \hat{k} \right)\]
\[ = 12 \left( \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{2} \hat{j} - \frac{1}{2} \hat{k} \right)\]
\[ = 6 \left( \sqrt{2} \hat{i} + \hat{j} - \hat{k} \right)\]