[Physics Class Notes] on Properties of Vector Pdf for Exam

In Physics terminology, you must have heard about scalar and vector quantities. We often define any physical quantity by magnitude. Hence the physical quantity featured by magnitude is called a scalar quantity. That’s it! But there are also physical quantities that have a certain specific magnitude along with the direction. Such a physical quantity represented by its magnitude and direction is called a vector quantity. Thus, by definition, the vector is a quantity characterized by magnitude and direction. Force, linear momentum, velocity, weight, etc. are typical examples of a vector quantity. Unlike scalar quantity, there is a whole lot to learn about vector quantity.

Before learning about the vector quantities and their properties, let us differentiate between the scalar quantities and vector quantities.

Scalar Quantities

• These are the quantities that have only magnitude and no direction. 

• The scalar quantities are one-dimensional.

• With the change in magnitude, scalar quantities also change.

• The mathematical operation is done between the two or more scalar quantities results in a scalar quantity. 

• Simple alphabets are used to denote scalar quantities.

• Example – speed, time, mass, volume, etc.

Vector Quantities

• These quantities have both magnitude and direction.

• They can be one, two or three-dimensional. 

• The vector quantities change when both magnitude and direction are changed.

• The resultant of two or more vector quantities is a vector quantity when mathematical operations are applied.

• To denote these quantities, an arrowhead is made above the alphabets.

• Example – displacement, force, velocity, acceleration, etc. 

Vectors are denoted by an arrow marked over a signifying symbol. For example,

[overrightarrow{a}] or [ overrightarrow{b}] [ overrightarrow{b}]

The magnitude of the vector [overrightarrow{a}] and [overrightarrow{b}] is denoted by ∥a∥ and ∥b∥ , respectively. 

Examples of the vector are force, velocity, etc. Let’s see below how it is represented 

Velocity vector:

[overrightarrow{v}]

Force vector:

[overrightarrow{F}]

Linear momentum:

[overrightarrow{p}]

Acceleration vector:

[overrightarrow{a}]

Force is a vector because the force is the magnitude of intensity or strength applied in some direction. Velocity is the vector where its speed is the magnitude in which an object moves in a particular path.

Classification Of Vectors 

There are various types of vectors that are used in Physics and Mathematics. Beneath are the names and descriptions of these vectors:

• Zero Vector – It is the type of vector whose magnitude is equal to zero.

• Position Vector – The vector which describes the position of a point in a cartesian system with respect to the origin is known as the position vector. 

• Unit Vector – The magnitude of this vector is equal to unity.

• Like Vectors – Like vectors are the vectors having the same direction.

• Unlike Vectors – These are the vectors having opposite directions.

• Co-initial Vectors – The co-initial vectors have the same starting point.

• Equal Vectors – The vectors which have the same magnitude, as well as the direction, are said to be equal vectors.

• Coplanar Vectors – Coplanar vectors are the vectors that are parallel to the same plane or lie in the same plane.

• Collinear Vectors – These are the vectors that are parallel to each other irrespective of their magnitudes and direction are known as collinear vectors. 

• Negative of a Vector – The two vectors having the same magnitude but different directions (opposite direction) are said to be the negative vectors of each other. 

• Displacement Vector – It is the vector that represents the displacement of a point from one position to another.  

Two-Dimensional Vectors Depiction

Two- dimensionally vectors can be represented in two forms, i.e. geometric form, rectangular notation, and polar notation.

1. Geometric Depiction of Vectors

In regular simple words, a line with an arrow is a vector, where the length of the line is the magnitude of a vector, and the arrow points the direction of the vector. 

2. Rectangular Depiction

In this form, the vector is placed on the  x and y coordinate system as shown in the image 

The rectangular coordinate notation for this vector is 

[overrightarrow{v}] = (6,3). An alternate notation is the use of two-unit vectors î = (1,0) and ĵ = (0,1) so that v = 6î + 3ĵ.

3. Polar Depiction

In the polar notation, we specify the vector magnitude r, r≥0, and angle θ with the positive x-axis. 

Now we will read different vector properties detailed below.

Equality of Vectors

If you compare two vectors with the same magnitude and direction are equal vectors. Therefore, if you translate a vector to position without changing its direction or rotating, i.e. parallel translation, a vector does not change the original vector. Both the vectors before and after changing position are equal vectors. Nevertheless, it would be best if you remembered vectors of the same physical quantity should be compared together. For example, it would be practicable to equate the Force vector of 10 N in the positive x-axis and velocity vector of 10 m/s in the positive x-axis.

Vector Addition

Think of two vectors a and b, the
ir sum will be a + b. 

The image displays the sum of two vectors formed by placing the vectors head to tail. 

Vector addition follows two laws, i.e. Commutative law and associative law.

A. Commutative Law – the order

in which two vectors are added does not matter. This law is also referred to as parallelogram law. Consider a parallelogram, two adjacent edges denoted by a + b, and another duo of edges denoted by, b + a. Both the sums are equal, and the value is equal to the magnitude of diagonal of the parallelogram

Image display that parallelogram law that proves the addition of vector is independent of the order of vector, i.e. vector addition is commutative

B. Associative Law – the addition of three vectors is independent of the pair of vectors added first.

(a+b)+c=a+(b+c).

Vector Subtraction

First, understand the vector -a. It is the vector with an equal magnitude of a but in the opposite direction.

The image shows two vectors in the opposite direction but of equal magnitude.

Therefore, the subtraction of two vectors is defined as the addition of two vectors in the opposite direction.

x – y = x + (-y)

Vector Multiplication by a Scalar Number

Consider a vector [overrightarrow{a}] with magnitude ∥a∥ and a number ‘n’. If a is multiplied by n, then we receive a new vector b. Let us see. Vector [overrightarrow{b}]= n  [overrightarrow{a}]   The magnitude of the vector  [overrightarrow{b}]   is ∥na∥. 

The direction of the vector [overrightarrow{b}] is the same as that of the vector a [overrightarrow{a}]

If the vector [overrightarrow{a}] is in the positive x-direction, the vector b [overrightarrow{b}] will also point in the same direction, i.e. positive x-direction.

Suppose if we multiply a vector with a negative number n whose value is -1. Vector [overrightarrow{b}] will be in the opposite direction of the vector [overrightarrow{a}]

The Vector Product 

The cross product of two vectors is equal to the product of the magnitude of the two given vectors and sine of the angle between these vectors. The vector product is represented as 

A x B = |A| |B| sin θ nˆ

 

Where,

A and B are two vectors

|A| = magnitude of vector A

|B| = magnitude of vector B

θ = angle between the vectors A and B

 

[hat{n}] = unit vector perpendicular to the plane containing the two vectors

 

Some properties of the vector product are discussed below:

A x B ≠ B x A 

But, A x B = (-B) x A

A x (B + C) = A x B + A x C

• When the vectors are perpendicular to each other then the vector product is maximum. 

• Due to parallel and anti-parallel vectors, the cross product becomes zero.

• When a vector gets multiplied by itself, then it results in a zero vector. 

• The orthogonal unit vectors show the cross product in the following manner,

           i x i = j x j = k x k = 0

           i x j = k, j x k = i, k x i = j

           j x i = -k, k x j = -i, i x k = -j

 

Fun Facts

1. Do you know, scalar representation of vector quantities like velocity, weight is speed, and mass, respectively?

2. Scalar multiplication of vector fulfills many of the features of ordinary arithmetic multiplication like distributive laws

a(x + y) = xa + xb(a + b)y = ay + by 1x = x(−1)x = -x₀a = 0