Vector Addition and Vector Subtraction

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 3-D 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 non-zero 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.

triangle law of vector addition
triangle law of vector addition

In other words: If the tail-end of one vector be placed at the head or the arrow-end of the other, their sum or resultant vector R is drawn from the tail end of the first to the head-end 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.)

triangle law of vector addition
Triangular law of vector addition
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∣=✓(A2+B2+2ABcosθ)​

Now in Right angle triangle OBN, we have OB2=ON2+BN2

magnitude of vector addition

2. Direction of the Resultant vectors:

If vector R makes an angle α with vector A, then in OBN,

angle of vector addition

Parallelogram law of vectors addition

State parallelogram law of vector addition| What is parallelogram law of vector addition?

If two non-zero 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 R2 = ON2 + CN2

⇒ R2 = (OA + AN)2 + CN2

⇒ R2 = A2 + B2 + 2ABcosθ , so

parallelogram law of vector addition | vector subtraction
parallelogram law of vectors
 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 non-zero vectors are represented by the (n – 1) sides of an n-sided polygon taken in the same order then the resultant is given by the closing side of the nth side of the polygon taken in the opposite order. i.e.

polygon law of vector addition
polygon law of vectors
\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

Resultant of vector addition

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= ✓(A2 + B2 + 2ABcosθ)=✓(A2 + B2 + 2ABcos0)=✓(A2 + B2 + 2AB)= A+B

and the angle between vectors A & R will be,

tanα=Bsinθ/(A+Bcosθ) = 0

i.e. α = 0

i.e. Rmax=|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= ✓(A2 + B2 + 2ABcosθ)=✓(A2 + B2 + 2ABcos180)=✓(A2 + B2 – 2AB)= A-B

and the angle between vectors A & R will be,

tanα=Bsinθ/(A+Bcosθ) = 0

i.e. α = 0 or 180

i.e. Rmin=|A+B|min= A-B ⇒ 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= ✓(A2 + B2 + 2ABcosθ)=✓(A2 + B2 + 2ABcos90)=✓(A2 + B2)

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

vector addition | vector subtraction
subtraction of vectors

And the angle between vectors A and R is α2 given by

tan\alpha_{2}=\frac{Bsin\theta}{A-Bcos\theta}

Resolution of vector

resolution of vector piphysics.com
Resolution of vectors into Rectangular Components

Rectangular components of 3-D vectors

From fig.

rectangular components of 3-D vector
Rectangular components of 3-D vector

\vec{R}= \vec{R_{x}}+\vec{R_{y}}+\vec{R_{z}}

or 

\dpi{120} \fn_jvn \large \vec{R}=\vec{R_{x}}\hat{i}+\vec{R_{y}}\hat{j}+\vec{R_{z}}\hat{k}

If vector R makes an angle α with the x-axis, β with the y-axis, γ with the z-axis, then

vector direction cosines

where l, m, n are called Direction Cosines of vector R and

l2+ m2+ n2 = cos2α + cos2β + cos2γ =

\dpi{120} \fn_jvn \large \frac{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}} = 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

Rx = 4î , Ry = 0 and Rz = -3k̂

Hence the magnitude of vector R,

R = |R| = ✓(Rx2 + Ry2 + Rz2)

= ✓(42 + 02 + (-3)2) = 5

Direction cosines of vector R will be

l = Rx/R = 4/5

m = Ry/R = 0/5 = 0

n = Rz/R = -3/5

Example 2. If vectors A and B be respectively equal to (3 – 4 + 5) and (2 + 3 – 4) , obtain the magnitude of vector A+B and AB.

Solution: vector A+B = (3 – 4 + 5)+(2 + 3 – 4)

= 5 +

Now the magnitude of vector A+B,

|A+B| = ✓{52+(-1)2+12} = ✓(27)

And vector AB = (3 – 4 + 5)-(2 + 3 – 4)

= -7 +9

Now the magnitude of vector A+B,

| AB| = ✓{12+(-7)2+92} = ✓(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| = ✓(Rx2 + Ry2 )

=✓(32 + 42) = 5 newton.

Example 4. A force is represented by 2 + 3 +6k. What is the magnitude of force?

Solution: R = ✓(22 + 32+ 62) = ✓(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

Rsum =|P+Q| = ✓(P2 + Q2 + 2PQcosθ)=1

or, ✓(12 + 12 + 2.1.1.cosθ) = 1

or cosθ= -1/2

Now the magnitude of difference becomes

Rdiff= |P-Q| =✓(P2 + Q2 – 2PQcosθ)

= ✓{12 + 12 – 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 non-coplanar 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 anti-parallel.
  • 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 A2 + B2 = C2, 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 A1 + A2 + A3 + A4 ………. + An and A1 = A2 = A3 = A4…….= An, then the adjacent vector are inclined to each other at angle 2π / n .
Free Mock Test 1 : Vector Addition and Vector Subtraction(theory-based)

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Addition and Subtraction Vectors (Theory Based)

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  1. Complete Practice with 20+ important questions.
  2. Maximum questions are selected from previous exams.
  3. Firstly read the above paragraphs of the articles and then attempts.

1 / 21

1. Minimum number of non-coplaner vectors whose resultant can be zero is

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2. For the resultant of the two vectors to be minimum, what must be the angle between them

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3. The rectangular components cannot have magnitude .............than that of the vector itself.

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4. Unit vector generally represents

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5. For the resultant of the two vectors to be maximum, what must be the angle between them

6 / 21

6. Select the right equation for vectors a & 

vector subtraction R=b-a

7 / 21

7. Select the right option for vectors a b

subtraction of two vectors R=a-b

8 / 21

8. Select the right equation for vector addition of a and b

vector addition R=a+b

9 / 21

9. Minimum number of collinear vectors whose resultant can be zero is

10 / 21

10. A vector can have only ......... rectangular components in plane.

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11. Minimum number of coplaner vectors whose resultant can be zero is

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12. the resultant of two vectors of unequal magnitude can never be a..

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13. A vector can have how many maximum number of components ?

14 / 21

14. Select the right equation for vectors a & b

vector subtraction r=[b-a]

15 / 21

15. If vector A, B and C lie in one plane and

A + B = C then

16 / 21

16. Position vector of origin becomes

17 / 21

17. The displacement vector of a stationary body becomes

 

18 / 21

18. how many minimum number of components of a vector can be?

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19. select right equation

subtraction of vectors

 

20 / 21

20. A vector can have only ......... rectangular components in space.

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21. Is it always possible to add any two vectors?

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Free Mock Test 2 : Vector Addition and Vector Subtraction (numerical based)

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Addition and Subtraction Vectors (Numerical Based)

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Note:

  1. Complete Practice with 20+ important questions.
  2. Maximum questions are selected from previous exams.
  3. Firstly read the above paragraphs of the articles and then attempts.

1 / 21

1. If vectors  |AB| =| A| =| B|, the angle between vectors A and B is

2 / 21

2. Following sets of three forces act on a body. Whose resultant cannot be zero

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3. Maximum and minimum magnitudes of the resultant of two vectors of magnitudes P and Q are in the ratio 3 : 1. Which of the following relations is true

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4. The value of the sum of two vectors a and b with θ  as the angle between them is

  1. √(a² + b² + 2ab sinθ)
  2. √(a² + b² - 2ab cosθ)
  3. √(a² + b² + 2ab cosθ)
  4. none of these

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5. Five equal forces of 10 N each are applied at one point and all are lying in one plane. If the angles between them are equal, the resultant force will be

6 / 21

6.

There are two force vectors, one of 5 N and other of 12 N at what angle the two vectors be added to get resultant vector of 17 N, 7 N and 13 N respectively

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7. If vectors  |AB| =| A| =| B|, the angle between vectors A and B is

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8. Let the angle between two nonzero vectors and B  be 120° and resultant be vector C, then

  1.  vector must be equal to | A - B|
  2.  vector C must be less than | A - B|
  3.  vector C must be greater than | A - B|
  4.  vector C may be equal to | A - B|

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9. What vector must be added to the two vectors î - 2ĵ + 2k̂ and 2î + ĵ - k̂,  so that the resultant may be a unit vector along the x-axis

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10. The resultant of two vectors A and B is perpendicular to vector A and its magnitude is equal to half the magnitude of vector B. The angle between A and B is

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11. Forces F1 and F2 act on a point mass in two mutually perpendicular directions. The resultant force on the point mass will be

  1. F1 + F2
  2. F1 - F2
  3. √(F1² +F2²)
  4. F1² +F2²

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12. If a particle moves from point P (2,3,5) to point Q (3,4,5). Its displacement vector be

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13. For the resultant of the two vectors to be maximum, what must be the angle between them

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14. The vector projection of a vector (3î +4 k̂) on the y-axis is

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15. If vector A, B and C lie in one plane and

A + B = C then

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16. A particle is simultaneously acted by two forces equal to 4 N and 3N. The net force on the particle is

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17. If the sum of two unit vectors is a unit vector, then magnitude of difference is

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18. A force of 5 N acts on a particle along a direction making an angle of 60° with vertical. Its vertical component be 

19 / 21

19. For the resultant of the two vectors to be minimum, what must be the angle between them

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20. Two forces, each of magnitude F have a resultant of the same magnitude F. The angle between the two forces is

21 / 21

21. The resultant of vectors P and Q  is perpendicular to P. What is the angle between P and Q

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FAQs

  1. 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

  2. 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| = ✓(P2 + Q2 + 2PQcosθ)
    and angle α=tan-1[​Bsinθ​/(A+Bcosθ)]

  3. 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

  4. Is vector addition commutative?

    Yes, vector addition is commutative.
    i.e. a + b = b + a

  5. Is vector addition associative?

    Yes, vector addition is associative. i,e.
    ( a + b ) + c = a +( b +c )

  6. Is vector subtraction commutative?

    No, vector subtraction is not commutative.
    i.e. ab = a +(- b) ≠ (-b) + a

  7. How to find the resultant of two vectors?

    The resultant of two vectors is given by
    |R|=|P+Q| = ✓(P2 + Q2 + 2PQcosθ)

  8. How to find the resultant of two vectors?
    | The resultant of two vectors p and q is r.

    The resultant |r|=✓(p2 + q2 + 2pqcosθ)

  9. How many minimum numbers of vectors in different planes

    Two

  10. 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.

  11. What is triangle rule of vector addition?
    triangle law of vector addition

    If two non-zero 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.

  12. How do you do vector addition? | Addition of vectors formula Math

    The resultant, |R|=|P+Q| = ✓(P2 + Q2 + 2PQcosθ)
    and angle, α=tan-1[​Bsinθ​/(A+Bcosθ)]

  13. Maximum and Minimum magnitudes of the resultant of two vectors

    Maximum magnitudes of the resultant of two vectors
    Rmax=|A+B|max= A+B
    Minimum magnitudes of the resultant of two vectors
    ⇒ Rmin=|A+B|min= A-B

  14. 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)

  15. Can the component of a vector be greater than vector itself?

    No.

  16. How can the resultant of 13 vectors acting at a point be found?

    By using the polygon law of vectors

  17. The angle made by the vector A = + with the x-axis is

    45

Amit Patel

My name is Amit Patel. I'm a resident of Lohinda, a small town in Uttar Pradesh. I have done M.Sc (Physics) from K.N.I.P.S.S. Sultanpur. Now I am an educator and a blogger.

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