SCALAR QUANTITY
The physical quantities that have only magnitude but not direction are called scalar quantities. They can be multiplied, added, and subtracted according to the simple laws of algebra. Scalar quantities change with a change in magnitude only. E.g. mass, length, time, work, electric current, density, speed, etc.
VECTOR QUANTITY
The physical quantities that have both magnitude and direction are called vector quantities. A vector is represented by bold-faced letters or having arrow arrowhead i.e., (𝐀⃗). Vectors are added, subtracted, and multiplied according to the rules of vectors. For e.g. Displacement, velocity, Force, acceleration, momentum, etc.
Speed = 4 m/s (is a scalar) , velocity = 4 m/s towards north (is a vector)
Note: If a physical quantity is a vector, it has a direction, but the converse may or may not be true, i.e. if a physical quantity has a direction, it may or may not be a vector, such as time, pressure, surface tension, electric current etc. have directions but are not vectors because they don’t obey laws of vector addition.
Representation of vectors
A vector is represented by a straight line with an arrow. The tail is the starting point, the head is the ending point and the line measures the direction. The length of the line gives the value or magnitude of the vector.
Mathematically, vector is represented by 𝐀⃗. And sometimes it is represented by bold letter A.
The magnitude or modulus of 𝐀⃗ is written as | 𝐎𝐏⃗ | = | 𝐀⃗ | or simply A.
Graphical form:
e.g. if we want to represent a force of 5 N acting 45⁰ N of E
Magnitude of vector | 𝐀𝐁⃗ | = 5 N
TYPES OF VECTORS
Unit Vector: A vector having unit magnitude is called a unit vector. It is represented by a letter with a hat or cap over it. Mathematically, it can be written as,
3) Equal vector:
Two or more vectors having equal magnitude and the same direction are called equal vectors. The two vectors shown in fig. are equal vectors as they have both direction and magnitude equal i.e.
4) Negative vector
Two vectors having equal magnitude but opposite directions are called negative vectors. In other words, A negative vector is a vector with the same magnitude as another vector but pointing in the exact opposite direction.
5) Coplanar vector
Coplanar vectors are a set of two or more vectors that lie on the same plane in three-dimensional space or are parallel to the same plane. For three vectors to be coplanar, their scalar triple product must be zero, indicating they can be expressed as a linear combination of each other and do not extend beyond a single plane.
6) Collinear vector
Collinear vectors are two or more vectors that are parallel to the same line, irrespective of their magnitudes and direction. Two collinear vectors having the same direction (θ=0∘ ) are called like or parallel vectors. Two collinear vectors having opposite directions (θ) are called unlike or antiparallel vectors.
Note: Angle between collinear vectors is always 0° or 180°
7) Concurrent vector
The vectors acting at the same point are called concurrent vectors. In other words, A concurrent vector is a set of two or more vectors whose lines of action intersect at a common point.
Angle between two vectors:
Angle between two vectors means the 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)
Addition or composition of vectors:
Vectors don’t obey the ordinary laws of algebra because of direction. So, vectors are added geometrically. when two or more than two vectors are added then a single vector is obtained, which is called resultant vector.
The process of adding two or more vectors into a single vector (resultant vector) is called composition of vectors.
The rules for addition of vectors are as follows:
Adding two vectors: Triangle or Parallelogram law of vector addition
Adding more than two vectors: Polygon law of vector addition
The triangle law of vectors (Addition)
Parallelogram Law of Vector Addition
