Know About The Direction Cosines & Direction Ratios of a Line

Concepts of Direction Ratios of a Line in 3D Cartesian Coordinate System

A direction ratio is a measure of how a line in 3D space is oriented. There are three direction ratios: the x-direction ratio, the y-direction ratio, and the z-direction ratio. The x-direction ratio is the measure of how the line is oriented in the x-direction, the y-direction ratio is the measure of how the line is oriented in the y-direction, and the z-direction ratio is the measure of how the line is oriented in the z-direction.

The direction ratios are calculated by taking the cross product of the direction vectors of the line. The direction vectors are the vectors that point in the direction of the line. The x-direction vector is the vector that points in the x-direction, the y-direction vector is the vector that points in the y-direction, and the z-direction vector is the vector that points in the z-direction. The direction ratios are calculated by taking the cross product of the x-direction vector, the y-direction vector, and the z-direction vector.

The direction ratios are used to calculate the orientation of a line in 3D space. The orientation of a line can be used to determine the direction of a line in 3D space. The direction of a line in 3D space can be determined by taking the vector sum of the direction vectors of the line.

What is a Position Vector?

A position vector is a vector that identifies the location of a point in space. It is represented by an arrow that points from the origin of the coordinate system to the point in space. The position vector includes the x-coordinate, y-coordinate, and z-coordinate of the point.

Things to Remember Related to Direction Cosines

There are a few things to remember when working with direction cosines:

1. The direction cosines always sum to 1. This means that the vector represented by the direction cosines is always pointing in the direction of the original vector.

2. The direction cosines are always perpendicular to each other. This means that the vector represented by the direction cosines is always oriented in a right angle to the original vector.

3. The direction cosines always decrease in magnitude as the distance between the original vector and the vector represented by the direction cosines increases.

How to Calculate the Direction Cosines & Direction Ratios of a Line?

To calculate the direction cosines and direction ratios of a line, you need to know the coordinates of the line’s starting point and endpoint. The direction cosines are a set of three numbers that describe the direction of a line in terms of its x, y, and z coordinates. The direction ratios are a set of three numbers that describe the length of each side of the triangle formed by the line’s starting point, endpoint, and the z-axis.

Direction cosines when the line does not pass through the origin

You may wonder how the direction cosines are to be found when the line does not pass through the origin. The answer is simple. We consider another fictitious line parallel to our line such that the second line passes through the origin.

Now, the angles that this line makes with the three axes will be the same as that made by our original line and hence the direction cosines of the angles made by this fictitious line with the axes will be the same for our original line as well.

Derivation

Here, the line under question is labelled as OP. It passes through the origin and we are to find out the direction cosines of the line. Note that we will follow the three-dimensional Cartesian system to mark the coordinates of the point P (x, y, z).

Let us assume that the magnitude of the vector is ‘r’ and the vector makes angles α, β, γ with the coordinate axes. Now, using puthagoruds ’ theorem, we know that we can express the coordinates of the point P (x, y, z) 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

In the orthogonal system, we can represent r in its unit vector components form as –

r=xi^+yj^+zk^

Using the relations we established above, we can substitute the values of x, y, z to get the following –

r^=lri^+mrj^+nrk^

So, r^ = r/ | r | = li^+mj^+nk^

By interpreting the above statement, it can be said that the direction cosines are the coefficients of the unit vectors i^,j^,k^ when we express the unit vector r^ in terms of its rectangular components.