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.

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Solution

Suppose a vector $\vec{r}$ makes an angle 45 $°$ with $Ο Χ$ , 60 $°$ with $Ο Υ$ and having magnitude 12 units.
$l = \cos 45 ° = \frac{1}{\sqrt{2}} and m = \cos 60 ° = \frac{1}{2}$
$Now, l^{2} + m^{2} + n^{2} = 1 \Rightarrow \frac{1}{2} + \frac{1}{4} + n^{2} = 1 \Rightarrow n^{2} = \frac{1}{4} \Rightarrow n = - \frac{1}{2} [∵ The angle with the z - axis is obtuse]$
Therefore,
$\vec{r} = |\vec{r}| (l \hat{i} + m \hat{j} + n \hat{k}) = 12 (\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{2} \hat{j} - \frac{1}{2} \hat{k}) = 6 (\sqrt{2} \hat{i} + \hat{j} - \hat{k})$

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