Mathematics and Science were invented by humans to understand and describe the world around us. A lot of mathematical quantities are used in Physics to explain the concepts clearly. A few examples of these include force, speed, velocity, and work.
These quantities are often described as being a scalar or a vector quantity.
- Scalars and vectors are differentiated depending on their definition. A scalar quantity is defined as the physical quantity that has only magnitude, for example, mass and electric charge.
- On the other hand, a vector quantity is defined as the physical quantity that has both magnitudes as well as direction like force and weight.
- The other way of differentiating these two quantities is by using a notation. In this document, let us try to learn what is a vector and a scalar quantity.
What is a Scalar Quantity?
Scalar quantity is defined as the physical quantity with magnitude and no direction.
- A scalar quantity only has a magnitude and a unit, it can be represented by a number only. A scalar does not have any direction.
Example: Mass = 4 kg - The magnitude of mass = 4, Unit of mass = kg
- Scalar quantities can be added, subtracted, and multiplied by simple laws of algebra.
- Examples of Scalar Quantities: There are plenty of scalar quantity examples, some of the common examples are: Mass, Speed, Distance, Time, Area, Volume, Density & Temperature
Question for Vector & Scalar Quantities
Try yourself:Which of the following is a scalar quantity?
Explanation
Work has only a magnitude but no direction. Therefore, work is a scalar quantity.
What is a Vector Quantity?
A vector quantity is defined as the physical quantity that has both direction as well as magnitude.
- Vector is the physical quantities having magnitude as well as specified direction.
Specified Direction
- For example: 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 |A|.
- Example of Vector quantity: Displacement, velocity, acceleration, force, etc.
Table: Difference between Vector and Scalar
General Points Regarding Vectors
1. Representation of Vector- Geometrically, the vector is represented by a line with an arrow indicating the direction of the vector as:
Vector
- Mathematically, the vector is represented by . Sometimes it is represented by the bold letter A.
- Thus, the arrow in the above figure represents a vector in XY-plane making an angle θ with the x-axis.
- A representation of the 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 an arrow in the direction of the vector.
2. The Angle Between Two Vectors (θ)The 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.
3. Negative of VectorIt implies vectors of the same magnitude but opposite in direction.
Negative vector
4. Equality of VectorsVectors having equal magnitude and same direction are called equal vectors.
5. Collinear VectorsAny two vectors are collinear, then one can be expressed in terms of others. = (where λ is a constant)
Collinear vector
Question for Vector & Scalar Quantities
Try yourself:What is the magnitude of a unit vector?
Explanation
The magnitude of a unit vector is unity. It just gives the direction of a vector. A unit vector has no units or dimensions.
6. Co-initial VectorIf two or more vectors start from the same point, then they called co-initial vectors.
Co-initial vector
Here A, B, C, D are co-initial vectors.
7. Coplanar VectorsThree (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.
Coplanar Vectors
Note:
If the frame of reference is translated or rotated the vector does not change (though its components may change).
8. Multiplication and Division of a Vector by a Scalar- Multiplying a vector with a positive number λ gives a vector whose magnitude becomes 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 the multiplication of by
- At here and are colinear vector.
9. Multiplication of a Vector- By a Real Number
When a vector A is multiplied by a real number n, then its magnitude becomes n times but direction and unit remains unchanged. - By a Scalar
When a vector A is multiplied by a scalar S, then its magnitude becomes S times, and unit is the product of units of A and S but direction remains same as that of vector A.
10. Scalar or Dot Product of Two Vectors- The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them. It is denoted by . (dot)
A . B = AB cos θ - The scalar or dot product of two vectors is a scalar.
Properties of Scalar Product
- Scalar product is commutative, i.e.,
A . B= B . A - Scalar product is distributive, i.e.,
A . (B + C) = A . B + A . C - Scalar product of two perpendicular vectors is zero.
A . B = AB cos 90° = 0 - Scalar product of two parallel vectors is equal to the product of their magnitudes, i.e., A . B = AB cos 0° = AB
- Scalar product of a vector with itself is equal to the square of its magnitude, i.e.,
A . A = AA cos 0° = A^{2} - Scalar product of orthogonal unit vectors and Scalar product in cartesian coordinates
= A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}
Question for Vector & Scalar Quantities
Try yourself:Dot product of two mutual perpendicular vector is
Explanation
Dot product of two mutual perpendicular vector is as angle between them is 90º and A.B = ABcosθ
= AB cos90° = 0
11. Vector or Cross Product of Two Vectors- The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by X (cross).
A X B = AB sin θ n
Vector cross product
- The direction of unit vector n can be obtained from right hand thumb rule.
- If fingers of right hand are curled from A to B through smaller angle between them, then thumb will represent the direction of vector (A X B).
- The vector or cross product of two vectors is also a vector.
Properties of Vector Product
- Vector product is not commutative, i.e.,
A X B ≠ B X A [∴ (A X B) = - (B X A)] - Vector product is distributive, i.e.,
A X (B + C) = A X B + A X C - Vector product of two parallel vectors is zero, i.e.,
A X B = AB sin 0° = 0 - Vector product of any vector with itself is zero.
A X A = AA sin 0° = 0 - Vector product of orthogonal unit vectors.
- Vector product in cartesian coordinates.
Question for Vector & Scalar Quantities
Try yourself:The angle between the vectors (A x B) and (B x A) is:
Explanation
(A × B) = - (B × A) which are equal and opposite in direction.
Hence it will have an angle in between 180°
Direction of Vector Cross Product
- When C = A X B, the direction of C is at right angles to the plane containing the vectors A and B. The direction is determined by the right hand screw rule and right hand thumb rule.
- Right Hand Thumb Rule Curl the fingers of your right hand from A to B. Then, the direction of the erect thumb will point in the direction of A X B.
Right Hand Thumb Rule
- Right Hand Screw Rule Rotate a right handed screw from first vector (A) towards second vector (B). The direction in which the right handed screw moves gives the direction of vector (C).