How to Find Vector Direction and Solve Ratio Problems
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 Made Simple
1. What are the direction cosines (DCs) of a vector in 3D space?
Direction cosines are the cosines of the angles that a vector makes with the positive directions of the x, y, and z axes. If a vector makes angles ɑ, β, and γ with the x, y, and z axes respectively, its direction cosines (denoted as l, m, n) are:
- l = cos ɑ
- m = cos β
- n = cos γ
For any vector, the sum of the squares of its direction cosines is always equal to 1, i.e., l² + m² + n² = 1. They uniquely define the direction of a vector.
2. What are the direction ratios (DRs) of a vector and how are they related to direction cosines?
Direction ratios of a vector are any three numbers (let's call them a, b, c) that are proportional to its direction cosines (l, m, n). This means there is some constant k such that:
- a = kl
- b = km
- c = kn
Unlike direction cosines, direction ratios are not unique. For example, if (2, 3, 4) are the direction ratios of a vector, then so are (4, 6, 8) and (6, 9, 12).
3. What is the main difference between direction cosines and direction ratios?
The fundamental difference lies in their uniqueness and definition. Here’s a quick comparison:
- Uniqueness: For a given direction, the direction cosines are unique. However, the direction ratios are not unique; they are an infinite set of proportional numbers.
- Magnitude: The sum of the squares of direction cosines is always 1 (l² + m² + n² = 1). The sum of the squares of direction ratios is not equal to 1 unless they are the same as the direction cosines.
- Definition: Direction cosines are the actual cosines of the angles with the coordinate axes. Direction ratios are simply numbers proportional to these cosines.
4. How do you find the direction of a vector from its components?
To find the direction of a vector, you calculate its direction cosines. For a vector r = xi + yj + zk, follow these steps:
- First, find the magnitude of the vector: |r| = √(x² + y² + z²).
- Then, divide each component of the vector by its magnitude to get the direction cosines:
- l = x / |r|
- m = y / |r|
- n = z / |r|
The set of values {l, m, n} precisely defines the vector's orientation in 3D space.
5. If direction ratios are not unique, why are they so commonly used in vector algebra?
Direction ratios are frequently used due to their convenience, even though they aren't unique. The primary reasons are:
- Simplicity: They often allow us to work with simple integers (e.g., <2, -1, 3>) instead of the fractions or irrational numbers that can appear in direction cosines.
- Geometric Equations: They are sufficient for defining the direction of a line in 3D. The equation of a line can be easily expressed using direction ratios.
- Parallelism: If two lines are parallel, their direction ratios are proportional. This property makes it very easy to check for parallelism without calculating the exact direction cosines.
6. How can you find the direction cosines if you only know the direction ratios (a, b, c)?
If you have the direction ratios (a, b, c) of a vector, you can find the corresponding direction cosines (l, m, n) by dividing each ratio by the square root of the sum of their squares. The formulas are:
- l = ± a / √(a² + b² + c²)
- m = ± b / √(a² + b² + c²)
- n = ± c / √(a² + b² + c²)
The choice of the plus or minus sign depends on the specific direction of the vector along the line.
7. Can a vector be parallel to one of the coordinate axes? If so, what would its direction cosines be?
Yes, a vector can be parallel to a coordinate axis. The direction cosines clearly show this:
- Parallel to the x-axis: The vector makes an angle of 0° with the x-axis and 90° with the y and z axes. Thus, its direction cosines are (cos 0°, cos 90°, cos 90°) = (1, 0, 0).
- Parallel to the y-axis: Similarly, its direction cosines are (0, 1, 0).
- Parallel to the z-axis: Its direction cosines are (0, 0, 1).
8. What are the direction cosines of a line that is equally inclined to all three coordinate axes?
If a line is equally inclined to the axes, it means the angles it makes are equal (ɑ = β = γ). Consequently, its direction cosines are also equal (l = m = n). Using the fundamental identity l² + m² + n² = 1, we get:
l² + l² + l² = 1
3l² = 1
l = ± 1/√3
Therefore, the direction cosines of such a line are (1/√3, 1/√3, 1/√3) or (-1/√3, -1/√3, -1/√3).
