Relation Between the Direction Cosines of a Line

The cosines of the angles formed by a (directed) line with the positive directions of the coordinate axes are called direction cosines.

Consider the following line OL, which passes through the origin O. Allow OL to be angled away from the coordinate axes.

–l, –m, and –n are the direction cosines for the line LO (i.e., the directed line segment in the opposite direction to OL).

Direction cosines of a line

The direction cosines of a directed line L that does not pass through the origin are the same as those of a directed line parallel to L that does pass through the origin.

The direction ratios are only three real numbers a, b, and c that are proportional to l, m, and n.

When the line does not pass through the origin, the direction cosines.

We’re looking for the direction cosines of the line OP, which passes through the origin. To mark the coordinates of the point P, we shall use the three-dimensional Cartesian system (x, y, z).

Assume that the vector’s magnitude is ‘r,’ and that the vector forms angles of α, β, γ with the coordinate axes. Using Pythagoras’ theorem, we now know that the coordinates of the point P (x, y, z) may be expressed as

x = r. cos α
y = r. cos β
z = r. cos γ
r = {(x – 0)² + (y – 0)² + (z – 0)²}^1/2
r = (x² + y² + z²)^1/2

Now, as we stated earlier, we can replace cos α, cos β, cos γ with l, m, n respectively. Thus, we have –

x = lr

y = mr

z = nr

We can write r in its unit vector components form in the orthogonal system as

Using the above-mentioned relationships, we may substitute the values of x, y, and z to obtain the following –

When we represent the unit vector r in terms of its rectangular components, the direction cosines are the coefficients of the unit vectors i,j,k, according to the previous statement.

The Direction Cosines and Their Relationship

Let OP be any line with direction cosines l, m, and n that passes through the origin O.

P is the point with the coordinates (x, y, z), and OP = r is the point with the coordinates (x, y, z).

Then OP² = x² + y² + z² = r²            …. (1)

From P draw PA, PB, PC perpendicular on the coordinate axes, so that OA = x, OB = y, OC = z.

Also, ∠POA = α, ∠POB = β and ∠POC = γ.

From triangle AOP, l = cos α = x/r ⇒ x = lr

Similarly y = mr and z = nr

Hence from (1) r² (l² + m² + n²) = x² + y² + z² = r²

 ⇒ l² + m² + n² = 1

Conclusion

The direction cosines of a directed line segment are the cosines of the angles formed by that line segment with the coordinate axes.

If the angle formed by the line segment with the coordinate axis is α, β, and γ , then these angles are called direction angles, and the cosines of these angles are called line direction cosines. As a result, the direction cosines are, cos α, cos β and cos γ are generally indicated by l, m, and n.

cos α = l, cos β = m and cos γ = n

Direction Ratios: The direction ratios are three integers that are proportional to a line’s direction cosines. As a result, if the drs are a, b, and c, and the dcs are l, m, and n, we must have