Concepts of Direction Ratios of a Line in 3D Cartesian Coordinate System
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.
FAQs on Direction Cosines & Direction Ratios of a Line
1. What are direction cosines (DCs) and direction ratios (DRs) of a line in 3D geometry?
In three-dimensional geometry, the orientation of a straight line is defined by its direction cosines and direction ratios.
Direction Cosines (DCs): If a line makes angles α, β, and γ with the positive x, y, and z-axes respectively, then cos α, cos β, and cos γ are the direction cosines of the line. They are typically denoted by l, m, and n. A key property is that l² + m² + n² = 1.
Direction Ratios (DRs): These are any three numbers (a, b, c) that are proportional to the direction cosines (l, m, n) of the line. This means a/l = b/m = c/n = k, for some non-zero constant k. For more details, you can explore the explanation of Direction Cosines.
2. What is the fundamental difference between direction cosines and direction ratios?
The fundamental difference lies in their uniqueness and definition:
- Uniqueness: For any given line, there is only one unique set of direction cosines (l, m, n). However, there are infinitely many sets of direction ratios (a, b, c), as any non-zero scalar multiple (like 2a, 2b, 2c) will also represent the direction of the same line.
- Definition: Direction cosines are the actual cosines of the angles the line makes with the coordinate axes. Direction ratios are simply a set of three numbers that are proportional to these cosines.
- Magnitude: The sum of the squares of direction cosines is always 1 (l² + m² + n² = 1), whereas this is not necessarily true for direction ratios.
3. How do you find the direction cosines of a line if its direction ratios are known?
If the direction ratios (DRs) of a line are a, b, and c, you can find its direction cosines (DCs) l, m, and n using the following formulas:
- l = ± a / √(a² + b² + c²)
- m = ± b / √(a² + b² + c²)
- n = ± c / √(a² + b² + c²)
Essentially, you divide each direction ratio by the square root of the sum of the squares of all three direction ratios. The ± sign indicates that a line has two possible sets of direction cosines, corresponding to the two opposite directions of the line (e.g., l, m, n and -l, -m, -n).
4. How can we determine the direction ratios of a line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂)?
The direction ratios (a, b, c) of a line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are found by subtracting the coordinates of the initial point from the coordinates of the terminal point.
The direction ratios are given by:
- a = x₂ - x₁
- b = y₂ - y₁
- c = z₂ - z₁
These values represent the components of the vector directed from P to Q. You can explore worked examples in the NCERT Solutions for Class 12 Maths Chapter 11.
5. Why is the sum of the squares of the direction cosines of any line always equal to 1?
This property (l² + m² + n² = 1) stems from the relationship between direction cosines and unit vectors. Consider a line passing through the origin with direction cosines l, m, n. A point P on this line at a distance 'r' from the origin has coordinates (lr, mr, nr).
If we consider a unit vector (a vector with magnitude 1) along this line, its components along the x, y, and z axes will be precisely the direction cosines l, m, and n. So, the unit vector can be written as û = lî + mĵ + n k̂. The magnitude of a unit vector is, by definition, 1.
Calculating the magnitude:
|û| = √(l² + m² + n²)
Since |û| = 1, we have:
1 = √(l² + m² + n²)
Squaring both sides gives us the identity: l² + m² + n² = 1.
6. How are direction cosines related to a unit vector along a line?
The direction cosines of a line are the components of the unit vector in the direction of that line. If a line has direction cosines l, m, and n, then the unit vector û parallel to that line is given by:
û = lî + mĵ + n k̂
This provides a direct link between the geometric concept of a line's orientation in 3D space and the algebraic representation of a vector. This relationship is a core concept in Vector Algebra and is fundamental to solving problems involving lines and planes.
7. If two lines are perpendicular, what is the condition involving their direction cosines or direction ratios?
If two lines are perpendicular to each other, the dot product of their direction vectors is zero. This leads to a specific condition for their direction cosines and direction ratios.
- Using Direction Cosines: If two lines have direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂), they are perpendicular if and only if:
l₁l₂ + m₁m₂ + n₁n₂ = 0 - Using Direction Ratios: Similarly, if the two lines have direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂), the condition for perpendicularity is:
a₁a₂ + b₁b₂ + c₁c₂ = 0
This condition is crucial for finding the angle between two lines and solving many problems in three-dimensional geometry.
