Vectors and their Addition & Subtraction | Physics Class 11 - NEET PDF Download

It is obvious in scalar algebra, that 3+4 = 7. But if for instance we assume that 3 and 4 are vectors rather than scalars then 3+4 could be anything between -7 to +7. Sounds interesting ! This is the real power of vectors. In this chapter we will study vector algebra and motion in two dimensions. 

Vectors and their Representation

A quantity is said to be a vector if it satisfies the following conditions. 
(a) Obeys the law of parallelogram addition (discussed later in document). 
(b) Has a specified direction.
A vector is represented by a line with an arrowhead. The point O from which the arrow starts is called the tail or initial point, or origin of the vector. Point A, where the arrow ends, is called the tip or head or terminal point of the vector. A vector displaced parallel to itself remains unchanged.

Types of Vectors

• Zero vector or Null vector: A vector is said to be a Zero Vector when the magnitude of the vector is zero and the starting point and the endpoint of the vector is the same. For instance, PQ is a line segment the coordinates of the point P are the same as that of the point Q. A Zero vector is denoted by 0.  The zero vector doesn’t have any specific direction.
• Unit vector: A vector is said to be a unit vector when the magnitude of the vector is 1 unit in length. Suppose if x is a vector having a magnitude x then the unit vector is denoted by x̂ in the direction of the vector, and it has a magnitude equal to 1. Two unit vectors cannot be equal as they might have different directions.
• Co-initial vectors: A vector is said to be a co-initial vector when two or more vectors have the same starting point, for example, Vectors AB and AC are called co-initial vectors because they have the same starting point A.
• Like and unlike vectors: The vectors having the same directions are said to be like vectors whereas vectors having opposite directions are said to be unlike vectors.
  
• Equal vectors: Two vectors are said to be equal vectors when they have both direction and magnitude equal, even if they have different initial points.
• Negative of a vector: Suppose a vector is given with the same magnitude and direction, now if any vector with the same magnitude but the opposite direction is given then this vector is said to be negative of that vector. Consider two vectors a and b, such that they have the same magnitude but opposite in direction then these vectors can be written as a = – b

Laws of Addition and Subtraction of Vectors

Triangle rule of addition

(i) Keep vectors s.t. tail of one vector coincides with head of others.

(ii) Join tail of first to head of the other by a line with arrow at head of the second.

(iii) This new vector is the sum of two vectors. (also called resultants)

(i)  (ii)  (iii) 

Polygon Law of addition  

This law is used for adding more than two vectors. This is extension of triangle law of addition. We keep on arranging vectors s.t. tail of next vector lies on head of former.

When we connect the tail of first vector to head off last we get resultant of all the vectors.

 [Associative Law]

Parallelogram law of addition 

Steps : 

(i) Keep two vectors such that their tails coincide.

(ii) Draw parallel vectors to both of them considering both of them as sides of a parallelogram.

(iii) Then the diagonal drawn from the point where tails coincide represents the sum of two vectors, with its tail at point of coincidence of the two vectors.

(i)  (ii)  (iii) 

Note: and  thus  [Commutative Law]

Note: Angle between 2 vectors is the angle between their positive directions.

Suppose angle between these two vectors is θ, and 

(AD)² = (AE)² +(DE)²

= (AB + BE)² + (DE)²

= (a +b cosθ)² + (b sinθ)² 

= a² + b² cos²θ + 2ab cosθ + b² sin²θ

= a² + b² + 2ab cosθ

Thus, AD = 

or 

angle α with vector a is

tan α =  = 

Important points :

•  To a vector, only a vector of same type can be added that represents the same physical quantity and the resultant is also a vector of the same type.
•  As R = [A² + B² + 2AB cosθ]^1/2 so R will be maximum when, cosθ = max = 1,

               i.e., θ = 0º, i.e. vectors are like or parallel and R_max = A + B.

•   and angle between them θ then R = 
•   and angle between them π -θ then R = 
•  The resultant will be minimum if, cosθ = min = -1, i.e., θ = 180º, i.e. vectors are antiparallel and R_min = A -B.
•  If the vectors A and B are orthogonal, i.e., θ = 90º, 
•  As previously mentioned that the resultant of two vectors can have any value from (A -B) to (A + B)      depending on the angle between them and the magnitude of resultant decreases as q increases 0º to 180º.
•  Minimum number of unequal coplanar vectors whose sum can be zero is three.
•  The resultant of three non-coplanar vectors can never be zero, or minimum number of non coplanar vectors whose sum can be zero is four.

Subtraction of Vectors

Negative of a vector say  is a vector of the same magnitude as vector but pointing in a direction opposite to that of .

Thus,  can be written as  or  is really the vector addition of  and .

Suppose angle between two vectors and is θ. Then angle between and  will be 180° -θ as shown in figure.

Magnitude of  will be thus given by

S =  = 

or S =  ...(i)

For direction of  we will either calculate angle α or β, where,

tanα =  =  ...(ii)

or tanβ =  =  ...(iii)

Ex.3 Two vectors of 10 units & 5 units make an angle of 120° with each other. Find the magnitude & angle of resultant with vector of 10 unit magnitude. 

⇒ Sol.  = 

 ⇒ α = 30°

[Here shows what is angle between both vectors = 120° and not 60°]

Note: or can also be found by making triangles as shown in the figure. (a) and (b)

                  Or                         

Ex.4 Two vectors of equal magnitude 2 are at an angle of 60° to each other find magnitude of their sum & difference. 

Sol. 

Ex.5     Find  and  in the diagram shown in figure. Given A = 4 units and B = 3 units.

Sol. Addition : 

R = 

=  =  units

tanα =  =  = 0.472

a = tan^-1(0.472) = 25.3°

Thus, resultant of  and  is  units at angle 25.3° from  in the direction shown in figure.

Subtraction : S = 

=  = 

and tanθ = 

=  = 1.04

∴ α = tan^-1 (1.04) = 46.1°

Thus,  is  units at 46.1° from  in the direction shown in figure.

Position Vector 

Position vector is a vector that gives the position of a point with respect to the origin of the coordinate system. The magnitude of the position vector is the distance of the point P from the origin O. Vector OP is the position vector that gives the position of the particle with reference to O.

Resolution of Vectors

If   the resultant, then conversely  i.e. the vector  can be split up so that the vector sum of the split parts equals the original vector   If the split parts are mutually perpendicular then they are known as components of  and this process is known as resolution. 

The orthogonal component of any vector along another direction which is at an angular separation θ is the product of the magnitude of the vector and cosine of the angle between them (θ). 

In physics, resolution gives unique and mutually independent components only if the resolved components are mutually perpendicular to each other. Such a resolution is known as rectangular or orthogonal resolution and the components are called rectangular or orthogonal components.

Results related to Vector Addition

Let P and Q be two vectors acting simultaneously at a point and represented both in magnitude and direction by two adjacent sides OA and OD of a parallelogram OABD as shown in figure. Let θ be the angle between P and Q and R be the resultant vector. Then, according to parallelogram law of vector addition, diagonal OB represents the resultant of P and Q

So, we have
R = P + Q
Now, expand A to C and draw BC perpendicular to OC.
From triangle OCB,

In triangle ABC,

Also,

Magnitude of resultant:

Substituting value of AC and BC in (i), we getwhich is the magnitude of the resultant.

Direction of resultant: Let ø be the angle made by resultant R with P. Then, From triangle OBC,

Products of Vectors

There are two kinds of multiplication for vectors. One kind of multiplication is the scalar product, also known as the dot product. The other kind of multiplication is the vector product, also known as the cross product. The scalar product of vectors is a number (scalar). The vector product of vectors is a vector.
The scalar product is discussed in detail in the document: Scalar Product of Vector.
The scalar product is discussed in detail in the document: Cross Product of Vector.