Addition and Substraction of Vectors

Vector Addition:Triangle, Parallelogram and Polygon Law of Vectors

As already discussed, vectors cannot be added algebraically. Following are a few points to remember while adding vectors:

Vectors are added geometrically and not algebraically.
Vectors whose resultant have to be calculated behave independently.
Vector Addition is nothing but finding the resultant of a number of vectors acting on a body.
Vector Addition is commutative. This means that the resultant vector is independent of the order of vectors.

Triangle Law of Vector Addition

The vector addition is done based on the Triangle law. Let us see what the triangle law of vector addition is:

Suppose there are two vectors, a and b.

Draw a line AB representing vector a with A as the tail and B as the head. Draw another line BC representing vector b with B as the tail and C as the head. Now join the line AC with A as the tail and C as the head. The line AC represents the resultant sum of the vectors a and b.

Vector Addition

The line AC represents the resultant sum of the vectors a and b.

\begin{array}{l} \vec{a} + \vec{b} \end{array}

The magnitude of vectors a and b is:

\begin{array}{l} \sqrt{a^{2} + b^{2} + 2 a b c o s θ} \end{array}

Where,

a = magnitude of vector a

b = magnitude of vector b

θ = angle between vector a and b

Let the resultant make an angle of Φ with vector a, then:

\begin{array}{l} t a n ϕ = \frac{b s i n θ}{a + b c o s θ} \end{array}

Let us understand this using an example. Suppose two vectors have equal magnitude A, and they make an angle θ with each other. Now, to find the magnitude and direction of the resultant, we will use the formulas mentioned above.

Let the magnitude of the resultant vector be B.

\begin{array}{l} B = \sqrt{A^{2} + A^{2} + 2 A A c o s θ} = 2 A c o s \frac{θ}{2} \end{array}

Let’s say that the resultant vector makes an angle Ɵ with the first vector.

\begin{array}{l} t a n ϕ = \frac{A s i n θ}{A + A c o s θ} = t a n \frac{θ}{2} \end{array}

Or,

\begin{array}{l} θ = \frac{θ}{2} \end{array}

Parallelogram Law of Vector Addition

The vector addition may also be understood by the law of parallelogram. The law states, “If two vectors acting simultaneously at a point are represented in magnitude and direction by the two sides of a parallelogram drawn from a point, their resultant is given in magnitude and direction by the diagonal of the parallelogram passing through that point.”

According to this law, if two vectors, P and Q, are represented by two adjacent sides of a parallelogram pointing outwards, as shown in the figure below, then the diagonal drawn through the intersection of the two vectors represent the resultant.

Parallelogram Law of Vector Addition

In Parallelogram Law of Vector Addition, the magnitude of the resultant is given by:
Magnitude of Resultant Vector
The direction of the resultant vector is determined as follows:

\begin{array}{l} \tan ϕ = \frac{C E}{A E} = \frac{Q \sin θ}{P + Q \cos θ} \end{array}

\begin{array}{l} θ = \tan^{- 1} [\frac{Q \sin θ}{P + Q \cos θ}] \end{array}

Polygon Law of Vector

According to the polygon law of vector addition, if the number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented by magnitude and direction such that the closing side of the polygon is taken in the opposite direction.
Let vector A, vector B, vector C and vector D be the four vectors for which the resultant has to be obtained.
Polygon law of vector addition
Consider triangle OKL, in which the vectors A and B are represented by sides OK, KL and are taken in the same order. Therefore, from the triangle law of vector addition, we know that the closing side OL is considered in the opposite direction such that it represents the resultant vector OR and KL.
Therefore,

\begin{array}{l} \vec{O K} + \vec{K L} = \vec{O L} . . e q .1 \end{array}

From the triangle law of vector addition, we know that triangle OLM can be expressed as vector OM is the resultant of the vectors OL and LM.

That is,

\begin{array}{l} \vec{O L} + \vec{L M} = \vec{O M} \end{array}

From eq.1,

\begin{array}{l} \vec{O K} + \vec{K L} + \vec{L M} = \vec{O M} . . e q .2 \end{array}

Again, applying the triangle law of vector addition to triangle OMN,

\begin{array}{l} \vec{O M} + \vec{M N} + \vec{O N} \end{array}

From eq.2, we get,

\begin{array}{l} \vec{O K} + \vec{K L} \end{array}

\begin{array}{l} \vec{L M} + \vec{M N} = \vec{O N} . . e q .3 \end{array}

Therefore,

\begin{array}{l} \vec{O K} = \vec{A} \end{array}

\begin{array}{l} \vec{K L}, \vec{B} \end{array}

\begin{array}{l} \vec{L M}, \vec{C} \end{array}

\begin{array}{l} \vec{M N}, \vec{D} \end{array}

Considering vector ON=vector R, the equation becomes

\begin{array}{l} \vec{A} + \vec{B} + \vec{C} + \vec{D} = \vec{R} \end{array}

Click the below video to understand the application of the polygon law of vector adding:

Addition and Substraction of Vectors

Vector Addition:Triangle, Parallelogram and Polygon Law of Vectors

Triangle Law of Vector Addition

Parallelogram Law of Vector Addition

Polygon Law of Vector

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