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.

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.)

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

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 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

 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 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= ✓(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. 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= ✓(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. R_min=|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

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

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

Resolution of vector

Rectangular components of 3-D vectors

From fig.

or 

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

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

l²+ m²+ n² = cos²α + cos²β + cos²γ =

= 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_x² + R_y² + R_z²)

= ✓(4² + 0² + (-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| = ✓{5²+(-1)²+1²} = ✓(27)

And vector A–B = (3î – 4ĵ + 5k̂)-(2î + 3ĵ – 4k̂)

= î -7ĵ +9k̂

Now the magnitude of vector A+B,

| A–B| = ✓{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| = ✓(R_x² + R_y² )

=✓(3² + 4²) = 5 newton.

Example 4. A force is represented by 2î + 3ĵ +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

R_sum =|P+Q| = ✓(P² + Q² + 2PQcosθ)=1

or, ✓(1² + 1² + 2.1.1.cosθ) = 1

or cosθ= -1/2

Now the magnitude of difference becomes

R_diff= |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 ………. + A_n and A₁ = A₂ = A₃ = A₄…….= A_n, 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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Free Mock Test 2 : Vector Addition and Vector Subtraction (numerical based)

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FAQs