Question

Relation between the Direction Cosines of a Line of three dimensional geometry.

Answer 1

Consider a line PQ with direction cosines l, m, n. Through the origin draw a line parallel to the given line and take a point R(x, y, z) on this line. From R draw a perpendicular RM on the x-axis.

Let OR = r

In right-angled triangle OMR,

∠OMR = 90° and ∠ROM = a

∴ cos α = \(\frac{OM}{OP} = \frac{x}{r}\)

⇒ l = \(\frac{x}{r}\)

⇒ x = lr

Similarly, y = mr and z = nr

Now, OR² = x² + y² + z²

⇒ r² = (lr)² + (mr)² + (nr)²

⇒ r² = r² (l² + m² + n²)

⇒ 1 = l² + m² + n²

Thus, l² + m² + n² = 1