How to Find Direction Ratios and Direction Cosines of a Vector with Formula and Examples
Every student has studied the concept of vectors in the middle school itself. However, it was usually related to the concepts of physics. By definition, a vector is defined as the quantity which has both magnitudes as well as direction. Some examples of vector quantities are velocity, displacement, acceleration, weight, momentum, force, etc.
In maths, the students are taught concepts of vector algebra and their operations, which help them solve questions related to their values and the other concepts. The key topics covered in the Vector Algebra chapter of Class 12 include:
Basic concepts.
Types of vectors.
Addition of vectores.
Direction cosines and ratios.
Basic Concepts of Vector Algebra
Position Vector
Consider a three-dimensional plane with origin O(0, 0, 0) and a point P located on the plane with the coordinates (x, y, z). The position vector is given by:
\[OP = \sqrt{x^{2} + y^{2} + z^{2}}\]
Relationship of magnitude, direction cosines, and direction ratios.
Let the magnitude be r.
The direction ratios have the coordinates (a, b, c).
The direction cosines have the coordinates (l, m, n).
The relation among the direction cosines and direction ratios is as follows:
l = a/r, m = b/r, n = c/r.
Vector Types
The various types of vectors included in the Class 12 concepts are as follows:
Null Vector or Zero Vector
The magnitude of these vectors is 0, and their initial and final points coincide. They are denoted by 0(vector). It does not have a definite direction. The zero vector is represented by AA(vector), BB(vector).
Unit Vector
A vector that has a unit length is called a unit vector.
Collinear Vectors
These are defined as those vectors parallel to the same line, irrespective of the direction and magnitude they hold.
Coinitial Vectors
The vectors which have the same initial point are called coinitial vectors.
Equal Vectors
Vectors having the same magnitude and same direction are said to be equal irrespective of their initial positions.
Negative Vectors
The vectors which are equal in magnitude but have opposite directions are called negative vectors.
Direction Cosines
Once you are well acquainted with the formula of position vectors, you must know the meaning of direction coines and direction cosines formula to solve the problems.
In the given figure, the position vector OP makes positive angles, namely, ⍺, 𝛽, and 𝛾 with the x-axis, y-axis and z-axis, respectively. The angles so formed are called direction angles. The direction cosines of a vector can be found out by taking the cosines of the above-mentioned angles.
Therefore, the direction cosines in a plane are given by the formulae:
cos ⍺ = x/r(vector)
cos 𝛽 = y/r(vector)
cos 𝛾 = z/r(vector)
We can rewrite the above equations in the form:
\[cos \alpha = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[cos \beta = \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[cos \gamma = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
Direction Ratios
The direction ratios of a vector are given as a product of magnitude and cosines of the given vector. These are given by the direction cosine formula of:
a = lr
b = mr
c = nr
The above mentioned direction ratios are proportional to the direction cosines. These can be expressed by the formulae:
\[l/a = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[m/b = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
\[n/c = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\]
Conclusion
At Vedantu, we provide you with comprehensive notes of direction ratios and cosines, allowing you to revise the concepts quickly and easily. Moreover, we provide you with the option of downloading these notes in PDF format, making the study process seamless. So, if you are preparing for your board or competitive examinations, you should be thorough with these concepts. Alongside, vector algebra forms an integral part of coordinate geometry in mathematics. This makes it quintessential for students to learn and understand the formulae and concepts well.
FAQs on Vector Direction And Ratios in Coordinate Geometry
1. What is vector direction and how is it represented?
Vector direction is the orientation of a vector in space, usually measured as an angle from a reference axis such as the positive x-axis. In two dimensions, direction is commonly expressed as:
- An angle θ measured anticlockwise from the positive x-axis.
- A direction ratio or component form like (a, b).
- A unit vector showing direction only.
2. What are direction ratios of a vector?
Direction ratios are the components of a vector that indicate its direction in space. If a vector is written as ai + bj + ck, then (a, b, c) are its direction ratios.
- They are proportional to the vector’s components.
- They do not need to be of unit length.
- Multiplying all ratios by the same number does not change the direction.
3. How do you find the direction ratios of a vector?
The direction ratios of a vector are simply its component values. To find them:
- Write the vector in component form, e.g., v = ai + bj + ck.
- The numbers (a, b, c) are the direction ratios.
4. What is the difference between direction ratios and direction cosines?
Direction ratios are the components of a vector, while direction cosines are the cosines of the angles the vector makes with the coordinate axes.
- Direction ratios: (a, b, c)
- Direction cosines: (cos α, cos β, cos γ)
- Relation: direction cosines = direction ratios ÷ magnitude of vector
5. How do you find the direction of a vector in 2D?
The direction of a 2D vector is found using θ = tan⁻¹(y/x). Steps:
- Let the vector be (x, y).
- Compute θ = tan⁻¹(y/x).
- Adjust θ based on the quadrant.
6. How do you find a unit vector in the direction of a given vector?
A unit vector in the direction of a vector is found by dividing the vector by its magnitude. Formula:
- If v = (a, b, c), then |v| = √(a² + b² + c²).
- Unit vector = v / |v|.
7. How are direction ratios related to the equation of a line?
Direction ratios determine the direction of a line in vector and Cartesian form. In 3D, the symmetric equation of a line is:
- (x − x₁)/a = (y − y₁)/b = (z − z₁)/c
8. Can direction ratios be negative?
Yes, direction ratios can be positive or negative depending on the vector’s direction.
- A negative ratio means movement in the negative axis direction.
- The sign indicates orientation.
9. How do you check if two vectors have the same direction?
Two vectors have the same direction if their components are proportional. To check:
- Compare ratios: a₁/a₂ = b₁/b₂ = c₁/c₂.
- If equal, vectors are parallel and have the same direction.
10. What is the formula for the magnitude of a vector using direction ratios?
The magnitude of a vector with direction ratios (a, b, c) is |v| = √(a² + b² + c²).
- Square each component.
- Add them.
- Take the square root.
