We cover all about of Vector Addition and Vector Subtraction, Triangle law, Parallelogram law and Polygon law of vector addition, Examples, Mock tests, Important FAQs.
We also discuss the Resolution of vectors, Rectangular components of 3D vectors, and Graphically adding and subtracting vectors.
Vector Addition
What is Vector Addition  Vector addition rules  Addition of vectors in physics
Vectors have both magnitude and direction. Magnitude can be added using the rules of algebra but not direction. So for vector addition using the rules of geometry.
There are certain laws to find the sum of given vectors when they are inclined at a certain angle.
 Triangle law of vector addition
 Parallelogram law
 Polygon law
Triangular law of vector addition
State triangle law of vector addition  state and explain triangle law of vector addition
What is Triangle law of vector addition? If two nonzero vectors { vector A & vector B} are represented by the two sides of a triangle taken in the same order then the resultant {vector R} is given by the closing side of the triangle in the opposite order.
In other words: If the tailend of one vector be placed at the head or the arrowend of the other, their sum or resultant vector R is drawn from the tail end of the first to the headend of the other. i.e.
\vec{R}=\vec{A}+\vec{B} \Rightarrow because \Rightarrow\vec{OB}=\overrightarrow{OA}+\vec{AB}
1. Magnitude of the resultant vector
In triangle ABN (from bellow fig.)
cos\theta=\frac{AN}{AB}=\frac{AN}{B}\Rightarrow AN=Bcos\theta
sin\theta=\frac{BN}{AB}=\frac{BN}{B}\Rightarrow BN=Bsin\theta
cosθ=AB/AN=B/AN⇒AN=Bcosθ
⇒R=∣R∣=∣A+B∣=✓(A^{2}+B^{2}+2ABcosθ)
Now in Right angle triangle OBN, we have OB^{2}=ON^{2}+BN^{2}
2. Direction of the Resultant vectors:
If vector R makes an angle α with vector A, then in △OBN,
Parallelogram law of vectors addition
State parallelogram law of vector addition What is parallelogram law of vector addition?
If two nonzero vectors { vector A & vector B} are represented by the two adjacent sides of a parallelogram then the resultant {vector R} is given by the diagonal of the parallelogram passing through the point of intersection of the two vectors.
1. Magnitude of the resultant vector:
Since R^{2} = ON^{2} + CN^{2 }
⇒ R^{2} = (OA + AN)^{2} + CN^{2}
⇒ R^{2} = A^{2} + B^{2 }+ 2ABcosθ , so
R=\vec{R}= \vec{A}+\vec{B}=\sqrt{A^{2}+B^{2}+2ABcos\theta}
2. Direction of resultant vectors:
tan\beta=\frac{CN}{ON}=\frac{Bsin\theta}{A+Bcos\theta}
Polygon law of vector addition
If a number of nonzero vectors are represented by the (n – 1) sides of an nsided polygon taken in the same order then the resultant is given by the closing side of the n^{th} side of the polygon taken in the opposite order. i.e.
\vec{R}=\vec{A}+\vec{B}+\vec{C}+\vec{D}+\vec{E}
\vec{OE}=\vec{OA}+\vec{AB}+\vec{BC}+\vec{CD}+\vec{DE}
Vector addition formula
The resultant of two vectors
If the angle between vector A and resultant vector R is α then,
tan\alpha=\frac{Bsin\theta}{A+Bcos\theta}
Special Conditions :
1. When both vectors are in the Same direction/Parallel, i.e. θ = 0 then⇒ cosθ = cos0 = 1 & sinθ= sin0=0
Now we have the resultant,
R= ✓(A^{2} + B^{2} + 2ABcosθ)=✓(A^{2} + B^{2} + 2ABcos0)=✓(A^{2} + B^{2} + 2AB)= A+B
and the angle between vectors A & R will be,
tanα=Bsinθ/(A+Bcosθ) = 0
i.e. α = 0
i.e. R_{max}=A+B_{max}= A+B ⇒ Thus, the maximum value of the resultant of two vectors is equal to the sum of their magnitude.
2. When both vectors are in the Opposite direction/ AntiParallel, i.e. θ =180 then⇒ cosθ = cos180 = 1 & sinθ= sin180=0
Now we have the resultant,
R= ✓(A^{2} + B^{2} + 2ABcosθ)=✓(A^{2} + B^{2} + 2ABcos180)=✓(A^{2} + B^{2} – 2AB)= AB
and the angle between vectors A & R will be,
tanα=Bsinθ/(A+Bcosθ) = 0
i.e. α = 0 or 180
i.e. R_{min}=A+B_{min}= AB ⇒ Thus, the minimum value of the resultant of two vectors is equal to the difference of their magnitude.
3. When both vectors are at the Perpendicular direction, i.e. θ = 90 then⇒ cosθ = cos90 = 0 & sinθ= sin90=1
Now we have the resultant,
R= ✓(A^{2} + B^{2} + 2ABcosθ)=✓(A^{2} + B^{2} + 2ABcos90)=✓(A^{2} + B^{2})
and the angle between vectors A & R will be,
tanα=Bsinθ/(A+Bcosθ)=B/A
4. The minimum number of vectors of unequal magnitude whose resultant can be zero is three.
5. The minimum number of vectors of equal magnitude whose resultant can be zero is two.
Properties of Addition of Vector
1. Commutative Law: The addition of Vector holds commutative law, i.e.
\vec{a}+\vec{b}= \vec{b}+\vec{a}
which means the order of vectors to be added does not affect the result.
2. Associative Law: The addition of Vector holds associative Law, i.e.
(\vec{a}+\vec{b}) +\vec{c}=\vec{a}+(\vec{b}+\vec{c})
which means the sum of the three vectors a, B & c is independent of the order in which they are added.
Subtraction of vectors
Law of vector subtraction How to subtract vectors?
Since
\vec{A}\vec{B}=\vec{A}+(\vec{B}).... and
\vec{A}+\vec{B}=\sqrt{\vec{A}^2+\vec{B}^2 + 2ABcos\theta}
\Rightarrow\vec{A}\vec{B}=\sqrt{\vec{A}^2+\vec{B}^2 + 2ABcos(180\theta)}
we know that cos(180−θ)= sinθ, so the Resultant of vector R
And the angle between vectors A and R is α_{2} given by
tan\alpha_{2}=\frac{Bsin\theta}{ABcos\theta}
Resolution of vector
Rectangular components of 3D vectors
From fig.
or
If vector R makes an angle α with the xaxis, β with the yaxis, γ with the zaxis, then
where l, m, n are called Direction Cosines of vector R and
l^{2}+ m^{2}+ n^{2} = cos^{2}α + cos^{2}β + cos^{2}γ =
= 1
Solved Examples
Example 1. If vector A= î + 2ĵ – 2k̂ , B = 2î + ĵ + k̂ , and C = î – 3ĵ – 2k̂, find magnitude and direction cosines of the vector A+B+C.
Solution: Vector sum R = A + B + C
= î + 2ĵ – 2k̂ + 2î + ĵ + k̂ + î – 3ĵ – 2k̂
= 4î – 3k̂
∴ R_{x} = 4î , R_{y} = 0 and R_{z} = 3k̂
Hence the magnitude of vector R,
R = R = ✓(R_{x}^{2} + R_{y}^{2} + R_{z}^{2})
= ✓(4^{2} + 0^{2} + (3)^{2}) = 5
Direction cosines of vector R will be
l = R_{x}/R = 4/5
m = R_{y}/R = 0/5 = 0
n = R_{z}/R = 3/5
Example 2. If vectors A and B be respectively equal to (3î – 4ĵ + 5k̂) and (2î + 3ĵ – 4k̂) , obtain the magnitude of vector A+B and A–B.
Solution: vector A+B = (3î – 4ĵ + 5k̂)+(2î + 3ĵ – 4k̂)
= 5î – ĵ + k̂
Now the magnitude of vector A+B,
A+B = ✓{5^{2}+(1)^{2}+1^{2}} = ✓(27)
And vector A–B = (3î – 4ĵ + 5k̂)(2î + 3ĵ – 4k̂)
= î 7ĵ +9k̂
Now the magnitude of vector A+B,
 A–B = ✓{1^{2}+(7)^{2}+9^{2}} = ✓(131)
Example 3. The rectangular components of a force are 3 N and 4 N. What is the magnitude of the force?
Solution: we know that for rectangular components the magnitude is given by
R = R = ✓(R_{x}^{2} + R_{y}^{2} )
=✓(3^{2} + 4^{2}) = 5 newton.
Example 4. A force is represented by 2î + 3ĵ +6k. What is the magnitude of force?
Solution: R = ✓(2^{2} + 3^{2}+ 6^{2}) = ✓(49) = 7 units
Example 5. If the sum of two unit vectors is a unit vector, then find the magnitude of difference.
Solution: Let us consider two unit vectors are P & Q . then according to question
R_{sum} =P+Q = ✓(P^{2} + Q^{2} + 2PQcosθ)=1
or, ✓(1^{2} + 1^{2} + 2.1.1.cosθ) = 1
or cosθ= 1/2
Now the magnitude of difference becomes
R_{diff}= PQ =✓(P^{2} + Q^{2} – 2PQcosθ)
= ✓{1^{2} + 1^{2} – 2.1.1.(1/2)} = ✓(3)
Tips & Tricks about vector addition and subtraction
 The minimum number of collinear vectors whose resultant can be zero is two.
 The minimum number of coplanar vectors whose resultant is zero is three.
 The minimum number of noncoplanar vectors whose resultant is zero is four.
 The magnitude of rectangular components of a vector is always less than the magnitude of the vector
 Resultant of two vectors will be maximum when θ = 0° i.e. vectors are parallel.
 The resultant of two vectors will be minimum when θ = 180° i.e. vectors are antiparallel.
 The vectors î ĵ k̂ are equally inclined to the coordinate axes at an angle of 54.74 degrees.
 If vector A + B = C and A^{2} + B^{2} = C^{2}, then the angle between vectors A and B is 90°. Also A, B and C can have the following values.
 (i) A = 3, B = 4, C = 5
 (ii) A = 5, B = 12, C = 13
 (iii) A = 8, B = 15, C = 17.
 If vectors A_{1} + A_{2} + A_{3} + A_{4} ………. + A_{n} and A_{1} = A_{2} = A_{3} = A_{4}…….= A_{n}, then the adjacent vector are inclined to each other at angle 2π / n .
Free Mock Test 1 : Vector Addition and Vector Subtraction(theorybased)
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Free Mock Test 2 : Vector Addition and Vector Subtraction (numerical based)
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FAQs

What are vector laws?
Basically, there are two laws of vector
1. Vector Addition
a. Triangle law
b. Parralelogram law
c. Polygon law
2. Vector Multiplication
a. Scalar Product
b. Vector Product 
How do you add two vectors?  What is the Formula For the Addition of Vectors? How to add vectors with angles?
The addition of vectors P & Q is given as
R=P+Q = ✓(P^{2} + Q^{2} + 2PQcosθ)
and angle α=tan^{1}[Bsinθ/(A+Bcosθ)] 
What is the vector law of addition?
There are certain laws to find the sum of given vectors when they are inclined at a certain angle.
1. Triangle law
2. Parallelogram law
3. Polygon law 
Is vector addition commutative?
Yes, vector addition is commutative.
i.e. a + b = b + a 
Is vector addition associative?
Yes, vector addition is associative. i,e.
( a + b ) + c = a +( b +c ) 
Is vector subtraction commutative?
No, vector subtraction is not commutative.
i.e. a – b = a +( b) ≠ (b) + a 
How to find the resultant of two vectors?
The resultant of two vectors is given by
R=P+Q = ✓(P^{2} + Q^{2} + 2PQcosθ) 
How to find the resultant of two vectors?
 The resultant of two vectors p and q is r.The resultant r=✓(p^{2} + q^{2} + 2pqcosθ)

How many minimum numbers of vectors in different planes
Two

What is meant by vector addition?
Vectors have both magnitude and direction. Magnitude can be added using the rules of algebra but not the direction, so for resolving this problem vectors are added by Geometry rules.

What is triangle rule of vector addition?
If two nonzero vectors { vector A & vector B} are represented by the two sides of a triangle taken in the same order then the resultant {vector R} is given by the closing side of the triangle in the opposite order.

How do you do vector addition?  Addition of vectors formula Math
The resultant, R=P+Q = ✓(P^{2} + Q^{2} + 2PQcosθ)
and angle, α=tan^{1}[Bsinθ/(A+Bcosθ)] 
Maximum and Minimum magnitudes of the resultant of two vectors
Maximum magnitudes of the resultant of two vectors
⇒ R_{max}=A+B_{max}= A+B
Minimum magnitudes of the resultant of two vectors
⇒ R_{min}=A+B_{min}= AB 
Can you apply commutative and associative law to vector subtraction?
a +( b) ≠ (b) + a, so commutative law is not applicable to vector subtraction. But associative law is applicable.
(a + b) – c = a + (b – c) 
Can the component of a vector be greater than vector itself?
No.

How can the resultant of 13 vectors acting at a point be found?
By using the polygon law of vectors

The angle made by the vector A = î + ĵ with the xaxis is
45