Question

If the direction cosines of the line are \(\frac{1}{2},-\frac{\sqrt{3}}{2}\),x respectively, then find the value of x.

(a) 1

(b) 0

(c) \(\frac{\sqrt{3}}{2}\)

(d) \(\frac{1}{2}\)

The question was asked in a national level competition.

My doubt stems from Direction Cosines and Direction Ratios of a Line in portion Three Dimensional Geometry of Mathematics – Class 12

Answer 1

Correct answer is (b) 0

The explanation is: If the direction cosines of a line are l,m,n respectively, then

l^2+m^2+n^2=1

∴\(\frac{1}{2}^2+(\frac{\sqrt{3}}{2})^2+x^2=1\)

x^2=\(1-\frac{1}{4}-\frac{3}{4}\)

x^2=0

⇒x=0