Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Direction cosines are the cosines of the angles which a line makes with the positive coordinate axes in anticlockwise direction. They are represented by $⟨l,m,n⟩$ where $\mathrm{l},\mathrm{m},\mathrm{n}$ correspond to the $\mathrm{x}$-axis, $\mathrm{y}$-axis, and $\mathrm{z}$-axis respectively. One important property of direction cosines is that the sum of their squares is unity. i.e., $l^2+m^2+n^2=1$ If the line lies in the zox plane, it implies that the $y$-axis is normal to the line as the $y$-axis is normal to the ZOX plane. Which means the angle the line makes with the $y$-axis is $90^{\circ }$ or $\frac{\pi }{2}$ radians. Thus, $m=\cos \left(\frac{\pi }{2}\right)=0$ It is given that the line makes an angle of $30^{\circ }$ or $\frac{\pi }{6}$ radians with the positive z-axis. Thus, $n=\cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$ Now we use the property $l^2+m^2+n^2=1$ Substituting the values of $m$ and $n$, we get the following values of । $l=\pm \frac{1}{2}$ We get two values of I because the only information given is that the line makes an angle of $\frac{\pi }{6}$ with the z-axis, so it can make an angle of either $\frac{\pi }{3}$ or $\frac{2\pi }{3}$ with the $x$-axis, which correspond to the two values of I. Therefore, the direction cosines are $⟨l,m,n⟩=⟨\pm \frac{1}{2},0,\frac{\sqrt{3}}{2}⟩$