IEMDaily - Video Lecture Notes

Example

Vector notation in 1-D

A 1-D vector can have only a single direction.
An example of 1-D vector is:

\vec{A} = 2 \hat{i}

where,

2

is the magnitude and

\hat{i}

is the direction of

\vec{A}

Example

Vector notation in 2-D

A 2-D vector quantity has two components.
An example of 2-D vector is:

\vec{A} = 2 \hat{i} + 3 \hat{j}

where

\hat{i}

and

\hat{j}

are two components of the vector and

2

and

3

are magnitudes in the respective directions.

Definition

Rectangular coordinate system

Rectangular coordinate system is a coordinate system that specifies each point uniquely in a plane by a triplet of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The coordinates of a point are often represented as

x, y

and

z

. The point of meeting of the three axis is known as origin. A vector is represented in rectangular coordinate system as:

\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

where

A_{x}, A_{y}

and

A_{z}

are distances of the vector measured along the

x, y

and

z

axis respectively,

Definition

Magnitude of vector in rectangular coordinate system

Let a vector be defined as

\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

Then, its magnitude is given by:

| \vec{A} | = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}

Definition

Addition of Vectros in Rectangular Co-ordinate System

Let's take two vectors in rectangular co-ordinate system:

\vec{a} = a_{1} \hat{i} + a_{2} \hat{j}

\vec{b} = b_{1} \hat{i} + b_{2} \hat{j}

Addition of Vectors:

\vec{a} + \vec{b} = (a_{1} + b_{1}) \hat{i} + (a_{2} + b_{2}) \hat{j}

Definition

Calculating Unit vector

A unit vector is a vector whose magnitude is 1 represented by a lowercase letter with a hat.
For example,

\hat{i}

is a unit vector.

\begin{matrix} 1 \\ \sqrt{3} \end{matrix} \hat{i} + \begin{matrix} 1 \\ \sqrt{3} \end{matrix} \hat{j} + \begin{matrix} 1 \\ \sqrt{3} \end{matrix} \hat{k}

is also a unit vector since its magnitude is

\sqrt{{(\frac{1}{\sqrt{3}})}^{2} + {(\frac{1}{\sqrt{3}})}^{2} + {(\frac{1}{\sqrt{3}})}^{2}}

Definition

Product of a scalar and a vector

When a vector is multiplied by a scalar, its magnitude is multiplied by the direction of the scalar.
If a vector is defined as

\vec{A} = | A | ∠ A

where

∠ A

is defined as the angle with respect to a given fixed line.
Then,

k \vec{A} = k | A | ∠ (A \pm 180^{o})

k < 0

k \vec{A} = k | A | ∠ (A)

k > 0

Example

Product of vector with scalar

The product of a scalar with vector is always a vector.
Example:
The product of mass 'm' and acceleration

\vec{a}

gives rise to a vector quantity 'force'.

\vec{F} = m \vec{a}

The product of mass 'm' and velocity

\vec{v}

gives rise to a vector quantity 'momentum'.

\vec{p} = m \vec{v}

Example

Vector notation in 3-D

A 3-D vector quantity has three components.
An example of 3-D vector is:

\vec{A} = 2 \hat{i} + 3 \hat{j} + 5 \hat{j}

where

\hat{i}, \hat{j}

and

\hat{k}

are the three components (i.e. directions) of the vector and

2, 3

and

5

are magnitudes in the respective directions.

Formula

Dot product of two vectors

Dot (Scalar) product:
Dot product of two vectors it is the scalar quantity which is a product of the magnitude of vectors and the cosine of the angle between them.
Represented as