Lesson 7: Graphical Method of Addition of Vectors in Two Dimensions (2-D)
Video Lesson
Simulation
Lesson objective
Dear learners,
At the end of the lesson, you will be able to
- Define the term resultant vector.
- Explain the geometric method for addition and subtraction of vectors in a plane.
- Apply geometric method of addition of vectors to find resultant of vectors in two dimensions.
Brainstorming question
Is it possible to add two vector quantities such as displacement vectors the way we do with scalar quantities like mass?
Key Terms and Concepts
- Graphical Method
- Algebraic Method
- Triangle Law
- Parallelogram Law
- Polygon Law
A technique for adding vectors visually by drawing them to scale on a graph and arranging them tip-to-tail.
A method for adding vectors using their components and applying algebraic operations to combine them.
A rule stating that two vectors can be added by placing them tip-to-tail, with the resultant vector drawn from the start of the first to the end of the second.
A method of vector addition where two vectors are represented as adjacent sides of a parallelogram, with the resultant vector as the diagonal.
A rule for adding multiple vectors by connecting them tip-to-tail in sequence, forming a polygon, with the resultant vector from the start of the first to the end of the last vector.
Methods Of Vector Addition
There are two commonly used methods employed to determine the vector sum of two or more vectors. The two methods that we will discuss in this unit are
I. Graphical Method
II. Algebraic Method
Graphical Method
- Graphical method of vector addition gives visual understanding of vectors and it is commonly used in navigation
- Figure 2.5 represents a trip that starts at point A then to point B and ending at point C. You see that total distance traveled for the whole trip is 1.2 km + 2 km = 3.2 km, but what is the displacement from A to C? How do you get the resultant displacement given displacements from A to B and from B to C?
- In this section we will discuss Triangle law, Parallelogram law and Polygon law of addition of vectors
- When two or more vectors are added they must have the same units.
Figure 2.3 A vector diagram with scale of a trip in a park
Triangle Law of Vector Addition
- Triangle Law of Vector Addition is a mathematical concept that is used to find the sum of two vectors.
- This law is used to add two vectors when the first vector’s head is joined to the tail of the second vector and then joining the tail of the first vector to the head of the second vector to form a triangle, and hence obtain the resultant sum vector.
- That’s why the triangle law of vector addition is also called the head-to-tail method for the addition of vectors.
- Triangle law of vector addition is used to find the sum of two vectors when the head of the first vector is joined to the tail of the second vector.
- Magnitude of the resultant sum vector R: $R = \sqrt{(P^2 + 2PQ \cos \theta + Q^2)}$
- Direction of the resultant vector R: ϕ = tan-1[(Q sin θ)/(P + Q cos θ)]
Figure 2.4 Triangle law of addition of vectors: S1 and S2 are placed head to tail.
Examples
Two vectors A and B have magnitudes of 4 units and 9 units and make an angle of 30° with each other. Find the magnitude and direction of the resultant sum vector using the triangle law of vector addition formula.
Solution:
The formula for the resultant vector using the triangle law are:
|R| = √(A2 + B2 + 2AB cos θ)
ϕ = tan-1[(B sin θ)/(A + B cos θ)]
So, we have
|R| = √(A2 + B2 + 2AB cos θ)
= √(42 + 92 + 2 × 4 × 9 cos 30°)
= √(16 + 81 + 72 × √3/2)
= √(97 + 36√3)
= 12.623 units
The direction of R is given by,
ϕ = tan-1[(B sin θ)/(A + B cos θ)]
= tan-1[(9 sin 30°)/(4 + 9 cos 30°)]
= tan-1[(9 × 1/2)/(4 + 9 × √3/2)]
= tan-1[(4.5)/(11.8)]
= 20.87°
Answer: Hence, the magnitude of the resultant vector is 12.623 units and the direction is 20.87°, approximately.
Parallelogram law of vector addition
Parallelogram law of vector addition states that
If two vectors are considered to be the adjacent sides of a parallelogram, then the resultant of the two vectors is given by the vector that is diagonal passing through the point of contact of two vectors.
Figure 2.5 Parallelogram law of vector addition
Polygon law of vectors addition
The polygon law of vector addition states that if the sides of a polygon are taken in the same order to represent a number of vectors in magnitude and direction, then the resultant vector can be represented in magnitude and direction by the closing side of the polygon taken in the opposite order.
Figure 2. 6 Polygon law of vectors addition
Subtraction of Vectors
The vector subtraction of two vectors a and b is represented by $\overrightarrow{A}-\overrightarrow{B}$ and it is nothing but adding the negative of vector b to the vector a. i.e.,$\overrightarrow{A}-\overrightarrow{B}$ =$ \overrightarrow{A}$+ ($-\overrightarrow{B}$). Thus, subtraction of vectors involves the addition of vectors and the negative of a vector.
Figure 2.7 Subtraction of vectors