Addition and Subtraction of Vectors: Class 11 Notes

Before moving on to the graphical representations of vector addition and subtraction, let's review the definition and types.  

Definition of Vectors

We can define vectors by two characteristics. 

• Magnitude - The size or length of the vector, which is always positive.
• Direction - The orientation of the vector in space (or coordinates in 2D and 3D spaces), which we represent using angles and unit vectors.

Types of Vectors

• Resultant Vector - This is a single vector that shows the combined effect of two or more vectors.
• Unit Vector - This is a vector with magnitude equal to 1. This is used to specify direction in space. We usually represent unit vectors as

$\hat{i},\hat{j},\hat{k}$

• Zero/Null Vector - A vector with zero magnitude. Even the direction is undefined. A simple example is when the initial and final positions coincide. In this case, the displacement is a null vector. 

Vector addition combines two or more vectors to produce a resultant vector. This mathematical operation can be represented graphically by two methods.

One is the parallelogram law, and the other is the triangle law.

Parallelogram Law of Vector Addition and Subtraction

The law of parallelogram states that when we add two adjacent sides of a parallelogram, the resultant vector is the diagonal of the parallelogram. This resultant vector originates from the same point where the two adjacent sides meet. 

Triangle Law of Vector Addition and Subtraction

The triangle law applies to the two adjacent sides of a triangle, where the resultant vector originates from the same point as the intersecting point.  

 

Let's see how we can graphically represent vector addition using the two laws we learnt above. 

Graphical Representations of Vector Addition using Parallelogram and Triangle Laws

Let’s consider two vectors, $\vec{A}$  and $\vec{B}$ .

We will see how the triangle and parallelogram laws apply in this context.  

• Triangle Law Method: You can place the tail of $\vec{B}$  at the head of $\vec{A}$ . Then the resultant $\vec{R}=\vec{A}+\vec{B}$ is the vector from the tail of $\vec{A}$  to the head of $\vec{B}$ . This method in your NCERT textbook is also referred to as the Head-to-Tail Method. 

 

• Parallelogram Law: Let's place both vectors' tails at a common origin. That should form a parallelogram. The result here is the diagonal from the origin to the opposite vertex.

 

 

Properties of Vector Addition

• The result of the vector addition depends on the angle between the vectors.
• Vector addition is always commutative. Meaning,  $\vec{A}+\vec{B}=\vec{B}+\vec{A}$ . 
• Vector addition also follows the associative law. That is, (A+B) + C = A + (B+C)
• The maximum vector resultant occurs when vectors are parallel. The angle between the two vectors is 0 degree.  
• The minimum vector resultant comes when vectors are anti-parallel. The angle then would be 180 degree. 

We use vector subtraction to find the difference between two vectors. We can represent this as the addition of one vector to the negative of another. 
Same as above, let’s look at how we can use the triangle and parallelogram laws to subtract vectors. 
What we need to consider here. 

To find $\vec{A}-\vec{B}$ , compute $\vec{A}+(-\vec{B})$ , where $-\vec{B}$  has the same magnitude as $\vec{B}$  but opposite direction.

• Graphically, we can place $\vec{A}$  and $-\vec{B}$  tail-to-head (triangle law).
• Or, in a parallelogram, the difference is the diagonal from the head of $\vec{B}$  to the head of $\vec{A}$  resultant's magnitude and direction are measured geometrically.

Here is a graphical representation of vector subtraction using both these laws. 

 

Properties of Vector Subtraction

• You should remember that vector subtraction is non-commutative: $\vec{A}-\vec{B}\ne \vec{B}-\vec{A}$
• The negative vector always reverses direction.
• We typically use vector subtraction in relative motion problems.
• The difference vector points from the head of B to the head of A when both start from the same origin. 

 

Here are some applications where vector addition and subtraction using the graphical method are quite common. 

Physics Applications

1. We can easily determine the net force acting on various objects using vector addition and subtraction. 
2. Knowing the graphical methods with triangle and parallelogram laws helps in visualising how to merge velocities and accelerations for problem-solving
3. Understanding how electric and magnetic fields interact

Engineering Applications

1. With addition and subtraction of vectors, it’s easier to break down forces in trusses and frameworks
2. Planning paths and coordinating arm movements in areas of robotics can be graphically represented with the addition or subtraction of vectors. 
3. Calculating navigation and trajectory paths is easier when using the graphical methods for vector

Everyday Applications

1. For GPS navigation, we generally apply the same physics of displacement vectors to plan routes
2. Even in areas of game development, professionals implement character movement and physics engines
3. Vector addition and subtraction using the graphical method helps in analysing projectile motion in sports, such as ball games

 

When you are preparing for JEE Main or similar tests, these are types of exercises you can practice. The solutions also have an additional concept, which you can read later in our guide to vector addition using the analytical method.   

Example 1 

Two displacements $\vec{A}=30\mathrm{m}$  east and $\vec{B}=40\mathrm{m}$  north are added. Sketch the resultant using the triangle law and estimate its magnitude and direction (scale: $1\mathrm{c}\mathrm{m}=10\mathrm{m}$  ).

Solution: Draw $\vec{A}$ ( 3 cm east), then $\vec{B}$  ( 4 cm north) from its head. The resultant $\vec{R}$  is from the tail of $\vec{A}$  to the head of $\vec{B}$ .

Magnitude (measured): $\left|\vec{R}\right|\approx 5\mathrm{c}\mathrm{m}=50\mathrm{m}$ . 

Direction (measured): $\theta \approx {53}^{\circ }$ north of east. 

Analytical check: $\left|\vec{R}\right|=\sqrt[]{{30}^2+{40}^2}=50\mathrm{m},\mathrm{t}\mathrm{a}\mathrm{n}\theta =\frac{40}{30}\approx {53.1}^{\circ }$ .

Diagram:

Diagram Description: Shows $\vec{A}$  (blue) east, $\vec{B}$  (green) north from $\vec{A}$  's head, and $\vec{R}$  (red) from origin to $\vec{B}$  's head.

Example 2

A velocity ${\vec{v}}_1=20\mathrm{m}/\mathrm{s}$  east is subtracted from ${\vec{v}}_2=20\mathrm{m}/\mathrm{s}$ north. Sketch ${\vec{v}}_2-{\vec{v}}_1$ (scale: $1\mathrm{c}\mathrm{m}=5\mathrm{m}/\mathrm{s}$  ). 

Solution: Compute ${\vec{v}}_2-{\vec{v}}_1=$ as ${\vec{v}}_2+\left(-{\vec{v}}_1\right)$ . Draw ${\vec{v}}_2$  ( 4 cm north), then $-{\vec{v}}_1$  ( 4 cm west) from its head. Resultant is from tail of ${\vec{v}}_2$  to head of $-{\vec{v}}_1$ .

Magnitude: $\left|\vec{R}\right|\approx 5.6\mathrm{c}\mathrm{m}=28\mathrm{m}/\mathrm{s}$ .

Direction: $\theta \approx {45}^{\circ }$  north of west.

Analytical check: $\left|\vec{R}\right|=\sqrt[]{{20}^2+{20}^2}\approx 28.3\mathrm{m}/\mathrm{s},\theta ={45}^{\circ }$ .

Diagram:

Diagram Description: Shows ${\vec{v}}_2$  (blue) north, $-{\vec{v}}_1$  (green) west, and $\vec{R}$  (red) from origin to $-{\vec{v}}_1$ 's head.

Example 3

Add forces ${\vec{F}}_1=50\mathrm{N}$  at ${30}^{\circ }$ to horizontal and ${\vec{F}}_2=50\mathrm{N}$  at ${60}^{\circ }$  using parallelogram law (scale: $1\mathrm{c}\mathrm{m}=10\mathrm{N}$  ).

Solution: Draw ${\vec{F}}_1\left(5\mathrm{c}\mathrm{m}\right.$  at $\left.{30}^{\circ }\right)$ and ${\vec{F}}_2\left(5\mathrm{c}\mathrm{m}\right.$  at ${\vec{F}}_2\left(5\mathrm{c}\mathrm{m}\right.$  from a common origin, completing the parallelogram. The resultant is the diagonal.

Magnitude: $\left|\vec{R}\right|\approx 8.6\mathrm{c}\mathrm{m}=86\mathrm{N}$ .

Direction: $\theta \approx {45}^{\circ }$  to horizontal.

Analytical check: Angle between vectors $={30}^{\circ }$ , so $\left|\vec{R}\right|=\sqrt[]{{50}^2+{50}^2+2\cdot 50\cdot 50\mathrm{c}\mathrm{o}\mathrm{s}{30}^{\circ }}\approx 86.6\mathrm{N},\theta \approx {45}^{\circ }$ .

 

Diagram:

Diagram Description: Shows ${\vec{F}}_1$  (blue) at ${30}^{\circ },{\vec{F}}_2$  (green) at ${60}^{\circ }$ , a dashed parallelogram, and $\vec{R}$  (red) as the diagonal.

Stuck up before exams? Don't cram. Instead, go through these Physics chapter notes. Find topic-wise guides under each chapter too! 

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes

Some more Science stream notes for Class 11 CBSE to check.

NCERT Class 11 Notes for PCM

NCERT Class 11 Physics Notes