Direction Cosines

The direction cosines are values of the angles of the three cosines of a vector that are made with the coordinate axes. Equivalently, another way to think of direction cosines is to see them as the components in a correlation of the unit vector signalling in the same direction. They have useful applications for developing direction cosine matrices that represent one set of orthonormal basis vectors in respect of another set, or for representing a known vector on a different basis.

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Direction Ratios of a Line

The fundamental concepts of 3-D geometry contribute to direction cosines and direction ratios. Direction cosines of a line that passes through the origin forming the angles with the coordinate axes are nothing but the numbers that are proportional to the direction cosines.

Direction Cosines in Three-Dimensional Geometry

In 3-D geometry, we have 3 axes: namely, the x, y, and z-axis. Suppose that a line OP passes through the origin in a 3D space. Then, the line will form an angle each with the x, y and z-axis respectively.

The cosines of each of these angles that the line creates along the x-axis, y-axis, and z-axis respectively are known as the direction cosines of the line in three-dimensional geometry. Usually, it is customary to express these direction cosines using the respective letters l, m, and n.

However, you need to remember that these cosines can only be found once we have discovered the angles that the line forms with each of the axes. Also, it is interesting to know that if we reverse the direction of this line, the angles will surely alter.

As a result, the direction cosines i.e. the cosines of these angles will also not be similar once the direction of the line has been reversed. Let us now consider a little different situation where our line does not pass through the point of origin (0, 0, and 0).

Direction Cosines in Case a Line Does Not Pass Through The Origin

You might be thinking over how the direction cosines are to be identified given that the line does not pass through the origin. It is simple. We will have to take into consideration another fictitious (assumed) line parallel to our line in a way that the 2^nd line passes through the origin.

Now, the angles that this line forms with the three axes i.e. x-axis, y-axis, z-axis will be similar to that formed by our original line and thus the direction cosines of the angles created by this fictitious line with the axes will be similar for our original line too.

Key Takeaways Direction Cosines

• We already know that [l = cos α], [m = cos β] and [n = cos γ] and we are also aware  that -1 < cos x < 1 ∀ x ∈ R, so ‘l’, ‘m’ and ‘n’ are real numbers with values fluctuating between -1 to 1. Thus, the value of direction cosine’s ∈ [-1, 1].

• The direction cosine of x, y and z axes are [1, 0, 0], [0, 1, 0] and [0, 0, 1] respectively.

• The dc’s of a line parallel to any coordinate axis are equivalent to the dc of the corresponding axis.

• The three angles formed along the x-axis with the coordinate axis are 0 degrees, 90 degrees and 90 degrees. Thus, the direction cosine turns to cos 0^o, cos 90^o, and cos 90^o i.e. [1, 0, 0].

• In case the assigned line is reversed, then the dc will be cos [π – α], cos [π – β], cos [π – γ]. Thus, a line can have two sets of direction cosines as per its direction.

• The direction cosines are linked by the relation l² + m² + n² = 1.

• The direction cosines of two parallel lines are always similar to one another.

• Direction ratios are directly in proportion to direction cosines and thus for a given line, there can be innumerable many direction ratios.