Question

If a line makes angles ${90}^o$, ${60}^o$ and ${30}^o$ with the positive direction of x, y and z axis respectively find its direction cosines.

Answer 1

Let the line be $x\hat{i}+y\hat{j}+z\hat{k}$

makes ${90}^o$ with $x$ direction.

$(x\hat{i}+y\hat{j}+z\hat{k}).\widehat{(j})=\cos 90=0$

$\Rightarrow x=0$

makes ${60}^o$ with $y$ axsis

$(x\hat{i}+y\hat{j}+z\hat{k}).\widehat{\left(j\right)}=\cos 60=\frac{1}{2}$

$y=\frac{1}{2}$

makes ${30}^o$ with $z$ axis

$(x\hat{i}+y\hat{j}+z\hat{k}).\widehat{\left(k\right)}=\cos {30}^o=\frac{\sqrt{3}}{2}$

$z=\frac{\sqrt{3}}{2}$

The vector is $\frac{1}{2}\hat{j}+\frac{\sqrt{3}}{2}\hat{k}$

it directional cosines are $(0,\frac{1/2}{\sqrt{\frac{1}{4}+\frac{3}{4}}},\frac{\sqrt{3}/2}{\sqrt{\frac{1}{4}+\frac{3}{4}}})$

$\Rightarrow (0,\frac{1}{2},\frac{\sqrt{3}}{2})$