Question

If l, m, n are the direction cosines of a position vector \(\vec{a}\), then which of the following is true?

(a) l^2+m^2-n^2=0

(b) lmn=1

(c) l^2+m^2+n^2=1

(d) l^2 m^2+n^2=1

This question was posed to me during an online interview.

Enquiry is from Vector Algebra Basics in portion Vector Algebra of Mathematics – Class 12

Answer 1

Correct choice is (c) l^2+m^2+n^2=1

Explanation: Consider \(\vec{a}\) is the position vector of a point M(x,y,z) and  α, β, γ are the angles, made by the vector \(\vec{a}\) with the positive directions of x, y and z respectively. The cosines of the angles, cos⁡α, cos⁡β, cos⁡γ are the direction cosines of the vector \(\vec{a}\) denoted by l, m, n, then

cos^2⁡α+cos^2⁡β+cos^2⁡γ=1 i.e.l^2+m^2+n^2=1.