General rule for addition and subtraction:- It states that the vectors to be added are arranged in such a way so that the head of first vector coincides with the tail of second vector, whose head coincides with the tail of the third vector and so on, then the single vector drawn from the tail of the first vector to the head of the last vector to the last vector represented their resultant vector.
Vectors are quantities that have both magnitude and direction. Examples are displacement, velocity, force, and momentum. The addition and subtraction of vectors is vital in physics and mathematics as it forms the basis for analysing various physical phenomena.
Addition of Vectors
The addition of vectors combines two or more vectors to find a resultant vector. There are two methods to add vectors:
1.Triangle Law of Vector Addition
This method states that if two vectors are represented as two sides of a triangle in sequence (tail of one vector at the head of the other), their resultant is represented by the third side of the triangle taken in the opposite order.
Steps:
Place the tail of the second vector at the head of the first vector.
Draw a vector from the tail of the first vector to the head of the second vector. This vector represents the resultant.
For example, if two vectors A and B are added, the resultant is: R = A + B
Example: Consider A = 5 units east and B = 3 units north. The resultant vector can be calculated by drawing the vectors as two sides of a triangle. The magnitude of R can be determined using the Pythagorean theorem:
R2 = A2 + B2 = 52 + 32 units
2. Parallelogram Law of Vector Addition
This method states that if two vectors originate from the same point and are represented as two adjacent sides of a parallelogram, the diagonal passing through the same origin represents their resultant vector.
Steps:
Place the two vectors with a common origin.
Complete the parallelogram by drawing lines parallel to each vector.
The diagonal from the origin represents the resultant vector.
The magnitude of the resultant is given by:
R2 = A2 + B2 + 2 AB cosθ
where A and B are the magnitudes of the two vectors and θ is the angle between them.
The direction of the resultant can be calculated using:
Bsinθ
Tanϕ = ————–
A + B cosθ
where ϕ is the angle the resultant makes with one of the vectors.
Subtraction of Vectors
The subtraction of vectors involves finding the difference between two vectors. It can be represented as the addition of one vector and the negative of another vector.
Steps for Subtraction:
Reverse the direction of the vector to be subtracted. This creates its negative vector.
Add the first vector to this negative vector using the triangle or parallelogram law.
For example, if you need to subtract B from A , the result is:
R = A − B = A+ (−B )
Example: If A = 5 units east and B = 3 units west, then:
The negative of B is 3 units east.
Adding A and −B:
R = A+ (−B) = 5 + 3 = 8 units east
Properties of vector subtraction :- a. Vector subtraction does not follow commutative law.
Vector subtraction does not follow associative law.
Graphical Representation
Head-to-Tail Method: Used in the triangle law, where vectors are arranged sequentially.
Component Method: Break vectors into their horizontal (x) and vertical (y) components and then add or subtract these components.
Applications
In physics, vector addition and subtraction are essential for understanding motion in two or three dimensions.
Force analysis in mechanics often requires the addition of forces acting at different angles.
Key Points to Remember
1.Vectors follow the commutative property for addition:
A + B = B + A
2. Subtraction does not follow the commutative property:
A − B ≠ B − A
3.Always resolve vectors into components when dealing with complex problems.
Understanding these basics makes handling vectors much easier, helping in real-world problem-solving and advanced studies in physics.
A vector is a quantity that has both magnitude and direction, such as force, velocity, and displacement. In contrast, a scalar only has magnitude and no direction, such as mass, time, or temperature.
To add two vectors using the triangle law:
Place the tail of the second vector at the head of the first vector.
Draw the resultant vector from the tail of the first vector to the head of the second vector.
The resultant vector represents the combined effect of the two vectors.
The parallelogram law states that if two vectors originate from the same point and are represented as adjacent sides of a parallelogram, their resultant is represented by the diagonal passing through the same origin. This method is useful when the vectors originate from a common point and form a specific angle.
If two vectors are perpendicular:
Use the Pythagorean theorem to find the magnitude of the resultant: ∣R∣=A2+B2
The direction can be found using trigonometry: tanθ = BA