In physics we deal with two types of physical quantities one is scalar and other is vector. Each scalar quantity has a magnitude and a unit.
For example Mass = 4 kg
Magnitude of mass = 4
and unit of mass = kg
Example of scalar quantities: Mass, Speed, Distance etc.
Scalar quantities can be added, subtracted and multiplied by simple laws of algebra.
Fig: Scalars and vectors
Vector are the physical quantities having magnitude as well as specified direction.
Speed = 4 m/s (is a scalar)
Velocity = 4 m/s toward north (is a vector)
If someone wants to reach some location then it is not sufficient to provide information about the distance of that location it is also essential to tell him about the proper direction from the initial location to the destination.
The magnitude of a vector is the absolute value of a vector and is indicated by or A.
Example of vector quantity : Displacement, velocity, acceleration, force etc.
Knowledge of direction:
Fig: Specified Direction
3. General Points Regarding Vectors
3.1 Representation of vector:
Geometrically, the vector is represented by a line with an arrow indicating the direction of vector as:
Mathematically, vector is represented by .
Sometimes it is represented by bold letter A.
Thus, the arrow in the above figure represents a vector in xy-plane making an angle θ with x-axis.
A representation of vector will be complete if it gives us direction and magnitude.
Symbolic form: used to separate a vector quantity from scalar quantities (u, i, m)
Graphical form: A vector is represented by a directed straight line,having the magnitude and direction of the quantity represented by it.
Example: If we want to represent a force of 5 N acting 45° N of E
(i) We choose direction coordinates.
(ii) We choose a convenient scale like 1 cm ≡ 1 N
(iii) We draw a line of length equal in magnitude and in the direction of vector to the chosen quantity.
(iv) We put arrow in the direction of vector.
Magnitude of vector:
3.2 Angle between two Vectors (θ):
Angle between two vectors means smaller of the two angles between the vectors when they are placed tail to tail by displacing either of the vectors parallel to itself (i.e 0 £ q £ p).
Ex.1 Three vectors are shown in the figure. Find angle between (i) and , (ii) and , (iii) and .
Ans: To find the angle between two vectors we connect the tails of the two vectors. We can shift & such that tails of and are connected as shown in figure.
Now we can easily observe that angle between and is 60º, and is 15º and between and is 75º.
3.3 Negative of Vector:
It implies vector of same magnitude but opposite in direction.
3.4 Equality of Vectors:
Vectors having equal magnitude and same direction are called equal vectors
3.5 Collinear vectors:
Any two vectors are collinear then one can be expressed in the terms of others.
= (where λ is a constant)
3.6 Co-initial vector: If two or more vector start from the same point then they called co-initial vectors.
Here A, B, C, D are co-initial vectors.
3.7 Coplanar vectors:
Three (or more) vectors are called coplanar vectors if they lie in the same plane or are parallel to the same plane. Two (free) vectors are always coplanar.
3.8 Multiplication and division of a vector by a scalar:
Multiplying a vector with a positive number λ gives a vector whose magnitude become λ times but the direction is the same as that of .
Multiplying a vector by a negative number λ gives a vector whose direction is opposite to the direction of and whose magnitude is λ times .
The division of vector by a non-zero scalar 'm' is defined as multiplication of by
At here and are colinear vector
Ex.2 A physical quantity (m = 3 kg) is multiplied by a vector such that . Find the magnitude and direction of if
(i) = 3m/s2 East wards
(ii) = -4 m/s2 North wards
(i) East wards
= 9 N East wards
(ii) North wards
= -12 N North wards
= 12 N South wards