Direction cosines

Introduction

Two of the most fundamental concepts in three-dimensional geometry are direction cosines and the direction ratio of a vector. In the following sections, we’ll learn about the direction cosines of a line traveling through the origin and making angles with coordinate axes. 

The Cosine Function Is Used To Show The Direction

The letters x, y, and z indicate the three axes of three-dimensional geometry: x, y, and z. Assume that a line OP runs through the center of the universe in three-dimensional space. 

In three-dimensional geometry, lines that intersect the x, y, and z axes are referred to as the direction cosines of the line. They are defined as follows: To preserve consistency, the letters l, m, and n are often employed to denote these direction cosines.

Cosine angles may be determined only after the angle formed by the line with each of the triangle’s axes is established. It’s worth noticing that the angle changes when the line is reversed.

Consequently, when a line is reversed in direction, the cosines of certain angles shift direction. Consider the following scenario in which our line does not intersect the origin: (0,0,0).

We can say this in the following manner:

When a line does not pass through the origin, cosines are used to determine the direction.

When a line does not pass through the origin, the calculation of direction cosines may be challenging to some. The answer is straightforward. Consider another imaginary line parallel to ours that passes through the center of the universe.

It is now feasible to understand how an imaginary line with three axes would provide equal angles and direction cosines to our original line.

The letters OP designate this specific line (overprinting) where O is the origin and P is the coordinate of a point in 3D. The objective is to determine the direction cosines of the line that passes through our beginning point. It is critical to notice that the coordinates of the point P will be denoted using the three-dimensional Cartesian system (x, y, z).

Consider a vector that has a magnitude of ‘r’ and that it defines the angles ‘α’, ‘β’, and ‘γ’ relative to the coordinate axes. According to the Pythagorean theorem, the coordinates of point P (x, y, z) are as follows: 

x = r. cos α

y = r. cos β

z = r. cos γ

r2 = {(x – 0)2+ (y – 0)2 + (z – 0)2}

r2 = (x2+ y2 + z2)

As stated before, we can replace cos α, cos β, cos γ with l, m, n respectively. Thus, we have –

x = lr

y = mr

z = nr

How Do You Calculate An Angle Using The Law Of Cosines?

Angles may be calculated using cosine laws in the following manner:

1. The cosine law may be utilized as a reference angle for the initial computation.
2. Then, using the rule of cosines, repeat this procedure with a new angle in mind.
3. Now multiply the triangle’s sides by 180 degrees to get the final angle.