Direction cosines of a line

Direction cosines of a line

Let us consider a line OP of length 'r' and the coordinates of P be (x,y,z). If, the line OP makes an angle of α, β and γ with the positive x-axis, y-axis and z-axis respectively as shown in figure. Then, the direction cosine or simple cosine of angle of that line is given by cosα, cos β and cos γ.

Cos α is generally denoted by l.

Cos β is denoted by m.

Cos γ is denoted by n. 

Then, we know that, P(x,y,z) means to reach the point P, we need to go 'x' along x-axis, 'y' along y-axis and 'z' along z-axis.

Hence, we can say that,

Cos α = x/r [As shown in figure]

l = x/r

x = lr

Cos β = y/r

m = y/r

y = mr

Cos γ = z/r

n = z/r

z = nr

Then, we know by the distance formula,

OP² = x² + y² + z²

r² = (lr)² + (mr)² + (nr)²

1= l² + m² + n²