Direction Cosines and Direction Ratios of a Line in 3D Geometry

Vector analysis is a part of the advanced subjects of mathematics and physics. It is used in different concepts to calculate the angle of the components of a vector quantity plotted on a 3D Cartesian system. The direction cosines & direction ratios of a line are a bunch of concepts developed to analyze the vector components and their directions based on the three axes of the Cartesian system. It is a very crucial part of analytical geometry that finds exceptional uses in advanced physics subjects studied in the engineering subjects. To understand the concept of direction ratios of a line, continue reading this section.

What is a Position Vector?

We all know that a vector quantity has both magnitude and direction. In fact, the direction property of a vector quantity differentiates it from a scalar quantity. If we consider point A in a 3D system and O as the origin then OA is the magnitude of this quantity and the direction from O to A is the other property.

Upon segregating the components of the vector quantity based on the three axes X, Y, and Z as l, m, and n respectively, we can write down the direction cosines as:

X = l|\[\vec{r}\]|

Y = m|\[\vec{r}\]|

Z = n|\[\vec{r}\]|

The expressions l|\[\vec{r}\]|, m|\[\vec{r}\]|, and n|\[\vec{r}\]| are considered to the direction ratios and are expressed by a, b, and c respectively. A direction cosine can be calculated for a vector by dividing the coordinate of that vector by the length of the same vector. In fact, adding the square of all the direction cosines of a vector will be equal to 1.

Once you learn how to determine the direction cosines, you can easily understand the meaning of the terms used in the formula used to find the direction cosines of the line joining the points.

Things to Remember Related to Direction Cosines

When the line segment OP is extended in the 3D Cartesian coordinate system, we will have to consider the supplement of the direction angles mentioned in the diagram to calculate the direction cosines and direction ratios. When the same line is reversed, the vector will also signify the opposite direction. The calculation will then be adjusted based on the changes in the direction of the vector quantity.

In a similar context, when a vector line does not pass through the origin O, another line segment parallel to the vector line is drawn that passes through O. This simple adjustment is done for easy calculation of the direction ratios of a line. This line segment should be parallel and of the same length as the vector quantity plotted on the 3D Cartesian coordinate system. This ensures that the angles made by the vector will be similar to that of the constructed line passing through the origin. Hence, it will ease our calculations as we can use the same formula.

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

If you follow the elaboration of the concept, you can understand the following things.

If r is the magnitude of vector OP then,

X = rcos α

Y = rcos β

Z = rcos γ

From this equation, we can conclude:

r = √ {(x – 0)2 + (y – 0)2 + (z – 0)2}

This equation becomes:

r = √ (x2 + y2 + z2)

When the value of x, y, and z are introduced as per the previous concept then, the equation becomes:

X = rcos α = l|\[\vec{r}\]|

Y = rcos β = m|\[\vec{r}\]|

Z = rcos γ n|\[\vec{r}\]|

On further replacing using the unit vector concept and representing the value of ‘r’ with those components; we can conclude that the direction cosines of the respective angles of the vector OP are nothing but the coefficients of the same unit vector. This happens when the rectangular components of the unit vector are considered. These are the direction cosines and direction ratios of a vector plotted on a 3D Cartesian coordinate system.

In fact, when we square and add the direction cosines, we find the result as 1. This answer can be concluded when we replace the values of the components on every axis by the direction cosines and then square it. As per the Pythagorean representation of a 3D Cartesian coordinate system, the answer will come as 1. Here the trigonometric identities will also be used.

To understand these concepts properly, you need to draw and correlate every explanation given in this segment niftily. This step will help you grab the concepts well. You will be able to find the direction cosines of the line joining the points manually and will find the values of the components without misreading them.