Vectors (Mathematics & Physics)

January 26, 2023 May 26, 2023

Vector Definition

To define a vector both magnitude and direction are required. There are many uses for vectors in math and physics. Some vector quantity examples are given below.

e.g.- Displacement, velocity, acceleration

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Scalars

The difference between scalars and vectors. Scalar quantities do not need any direction to express. Some scalar quantity examples are given below.

e.g.- Distance, Time

How to find the Magnitude of a vector?

If \(\overrightarrow{a}\) is such that \(\overrightarrow{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) then magnitude of \(| \overrightarrow{a} |\)

\(|\overrightarrow{a}| = \sqrt {(a_1)^2 + (a_2)^2 + (a_3)^2}\)

Type of Vectors

Let's talk about some vectors and their properties. Some Important vectors are given below.

Null or Zero vector

The vector has 0 magnitudes and no direction. It denoted by \(\overrightarrow{0}\). Also, it is the additive identity of any vector.

\(\overrightarrow{a} + \overrightarrow{0} = \overrightarrow{a}\)

Unit vector

A vector whose magnitude is 1 is called a unit vector and it is denoted by \(\hat{a} \). It is also the multiplicative identity of the vector.

\(\hat{a} = \frac{\overrightarrow{a}}{|\overrightarrow{a}|}\)

Negative vector

If the magnitude is the same for two vectors and the direction is opposite then vectors are called negative vectors of each other.

e.g.- If vector A and vector B have the same magnitude but the direction of B is opposite to A then B is called the negative vector of B.

\(|\overrightarrow{A}| = |\overrightarrow{B}| \)

\(\overrightarrow{A} = -\overrightarrow{B}\)

Reciprocal vector

If A and B are two vectors of the same direction but the magnitude of A inverse of B then B is called the reciprocal vector of A.

\(\overrightarrow{A} = \frac{1}{\overrightarrow{B}}\)

Parallel vector

Parallel vectors are those vectors whose direction is the same or opposite and whose magnitude may differ from each other but the direction of the angle is the same. you can see both types of parallel vectors.

Free vector

A vector that does not undergo any change in its transfer from one place to another is called a free vector.

Localised Vector

The vector that belongs to a fixed point is called a localised vector.

Colinear vectors

The vectors which lie on the same line are called colinear vectors.

Co-initial vector

Vectors that have the same starting point are called co-initial vectors.

\(\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD}, \overrightarrow{AE} \)

Co-terminal vector

Vectors that have the same endpoint are called co-terminal vectors.

\(\overrightarrow{BA}, \overrightarrow{CA}, \overrightarrow{DA}, \overrightarrow{EA} \)

Equal vector

The vectors which have the same magnitude and direction are called equal vectors.

Addition of vector

1. Triangle rule

\(\overrightarrow{AC} = \overrightarrow{AB}+\overrightarrow{BC} \)

Note:-

\(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA} = 0 \)

2. Parallelogram rule

\(\overrightarrow{AC}=\overrightarrow{AD}+\overrightarrow{AB} \)

Properties of the sum of vector

1. Commutative law

\(\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{a} \)

2. Associative law

\(\overrightarrow{a}+(\overrightarrow{b}+\overrightarrow{c}) = (\overrightarrow{a}+\overrightarrow{b})+\overrightarrow{c} \)

3. Additive identity

\(\overrightarrow{a}+\overrightarrow{0}=\overrightarrow{0}+\overrightarrow{a} \)

4. Additive inverse

\(\overrightarrow{a}+\overrightarrow{-a}=\overrightarrow{0} \)

\(\overrightarrow{-a}\) is additive inverse of \(\overrightarrow{a}\).

5. Cancelation law

\(\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{c} \)

\(\overrightarrow{a}=\overrightarrow{c} \)

Position vector

In 2-D

the position vector of point P = \(\overrightarrow{OP}\)

\(\overrightarrow{OP} = \overrightarrow{OM}+\overrightarrow{MP} \)

\(\overrightarrow{OP} = x\hat{i}+y\hat{j} \)

In 3-D

\(\overrightarrow{OP} = x\hat{i}+y\hat{j}+z\hat{k} \)

Scalar product or Dot product

\(\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}| cos θ \)

\(cos θ = \frac {\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}\)

\(max (\overrightarrow{a}.\overrightarrow{b})=|\overrightarrow{a}||\overrightarrow{b}| \)

\(min (\overrightarrow{a}.\overrightarrow{b})= -|\overrightarrow{a}||\overrightarrow{b}| \)

Note:- \(\hat{i}.\hat{i}= 1.1 cos 0 = 1\)

\(\hat{j}.\hat{j} = 1\)

\(\hat{k}.\hat{k} = 1\)

\(\hat{i}.\hat{j}= 1.1 cos 90 = 0\)

\(\hat{k}.\hat{j} = 0\)

\(\hat{i}.\hat{k} = 0\)

Vector product or Cross product

\(\overrightarrow{a}✕\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}| sin θ\space \hat{n} \)

\(\overrightarrow{a}✕\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}| sin θ \)