Direction Cosines and Direction Ratios of a Line: The Three-Dimensional Cartesian Coordinate System (For CBSE, ICSE, IAS, NET, NRA 2022)

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Direction Cosines

When a directed line OP passing through the origin makes axis angles with the axis respectively with O as the reference, these angles are referred as the direction angles of the line and the cosine of these angles give us the direction cosines. These direction cosines are usually represented as l, m and n.

If we extend the line OP on the three-dimensional Cartesian coordinate system, then to figure out the direction cosines, we need to take the supplement of the direction angles. It is pretty obvious from this statement that on reversal of the line OP in opposite direction, the direction cosines of the line also get reversed. In a situation where the given line does not pass through the origin, a line parallel to the given line passing through the origin is drawn and in doing so, the angles remain same as the angles made by the original line. So, we have the same direction.

Line passing through the origin to figure out the direction angles and their cosines, we can consider the position vectors of the line OP.

If , then from the above figure 1, we can see that

Where denotes the magnitude of the vector and it is given by,

The cosines of direction angles are given by and these are denoted by respectively. Therefore, the above equations can be reframed as:

We can also represent r in terms of its unit vector components using the orthogonal system.

Substituting the values of x, y and z, we have

Therefore, we can say that cosines of direction angles of a vector are the coefficients of the unit vectors when the unit vector is resolved in terms of its rectangular components.

Any number proportional to the direction cosine is known as the direction ratio of a line. These direction numbers are represented by a, b and c.

Also, as

In simple terms,

On dividing the equation, by we have,

Using equations 1,2 and 3, we get

We can conclude that sum of the squares of the direction cosines of a line is 1.

From the above definition, we can say that

From these relations, we get

The ratio between the direction cosines and direction ratios of a line is given by

But we know that

From this, we can find that

The value of k can be chosen as positive or negative depending upon the direction of the directed line.

We can take any number of direction ratios by altering the value of k. Also check: How to check Direction Cosine and Direction ratio

We are clear on the concept of the direction ratios and cosines of a line. To investigate more about three-dimensional geometry, download learning app and keep learning.

Question

Find the direction cosine of the line that makes equal angles with each of the coordinate axes.

Solution

Let us assume that the given line makes angles α, β, γ with the coordinate axes. The direction cosines of the line are given by We know that

Therefore, we use the relation

So,

Since the line makes equal angles with the coordinate axes,

Thus,

Hence, we can conclude that the line making equal angles with the coordinate axes has the direction cosines .

Question 2

How do we use the law of cosines to find an angle?

Solution

Laws of the cosines to find out an angle in the following ways:

• First of all, use the taw of the Cosines for calculating one of the angles.
• Then, use the law of the Cosines for a second time to find out another angle.
• Finally, use angles of a triangle add to 180 degrees to find out the last angle.

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