# Properties of Vector

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## What is a Vector?

In the Physics terminology, you must have heard about scalar and vector quantities. We often define any physical quantity by a magnitude. Hence physical quantity featured by magnitude is called a scalar quantity. That’s it! But there are also physical quantities which have a certain specific magnitude along with the direction. Such 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.

Vectors are denoted by an arrow marked over a signifying symbol. For example, $\overrightarrow{a}$ or $\overrightarrow{b}$. 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.

### Two-Dimensional Vectors Depiction

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

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.

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.

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 the 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 vector 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.

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

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

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

1. 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

1. 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 $\overrightarrow{a}$ . If the vector $\overrightarrow{a}$ is in the positive x-direction, the vector $\overrightarrow{b}$ 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 $\overrightarrow{a}$.

### Fun Facts

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

2. Scalar multiplication of vector fulfils 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 = -x

• 0a = 0

Q1. What is a Unit Vector?

A1. Vector whose magnitude is 1 unit. Therefore, a unit vector is majorly used to denote the direction of vector quantities. In Cartesian coordinates, usually: î, ĵ, k̂ = unit vector in x, y, z-direction respectively

The position vector of any object can be signified in Cartesian coordinates as:

r = xî + yĵ + zk̂.

When direction and not magnitude  is the major interest for any vector quantity, then vectors are normalised to unit length magnitude. Any vector is the combination of sum of the unit vector and scalar coefficients. The unit vector in the x-axis,y-axis,z-axis direction is i, j, and k , respectively.

Q2. What is the Zero Vector?

A2. Zero vector with no direction is an exception to vectors having direction. As the name suggests, the zero vector is a vector of the zero magnitudes. Because of its zero magnitudes, the zero vector does not point in any direction. There can only be a single vector of zero magnitudes. It is denoted by 0 as the length or magnitude is zero.Hence we say the zero vector.