Standard Unit Vectors

Overview

**Standard Unit Vectors**

» $3$ orthogonal axes are represented with $3$ unit vectors

→ x-axis $i$

→ y-axis $j$

→ z-axis $k$

unit means "one"

A person walks $5$m east and then takes the following path

•
$3$m north

•
$4.2$m south

•
$\frac{3}{4}$ m north

At this end position, how far is the person away from the starting point *in the east direction* ?

The answer is '$5$m '– as the person moved $5$ meter east and then all his movements were in directions north and south.

*Any change in a direction affects the component along that direction only* and does not affect the components in the directions at $90}^{\circ$ to that direction.

Independence of Quantities along orthogonal directions: For a vector, changes along one axis affect only the component along that axis and do not affect the components along other axes, as the axes are orthogonal.

There are $3$ orthogonal components in 3D coordinate space and $3$ orthogonal axes are defined for that.

Along the three orthogonal axes, irreducible unit is defined as unit vectors $i$, $j$, and $k$.

*3D vector space is of 3 orthogonal axes, with standard unit vectors $i$, $j$, $k$.*

summary

**Standard Unit Vectors: ** 3D vector space has three orthogonal axes. Unit vectors along the axes are standard unit vectors and are represented with $i$, $j$, and $k$.

Outline

The outline of material to learn vector-algebra is as follows.

Note: Click here for detailed outline of vector-algebra.

• Introduction to Vectors

→ __Introducing Vectors__

→ __Representation of Vectors__

• Basic Properties of Vectors

→ __Magnitude of Vectors__

→ __Types of Vectors__

→ __Properties of Magnitude__

• Vectors & Coordinate Geometry

→ __Vectors & Coordinate Geometry__

→ __Position Vector of a point__

→ __Directional Cosine__

• Role of Direction in Vector Arithmetics

→ __Vector Arithmetics__

→ __Understanding Direction of Vectors__

• Vector Addition

→ __Vector Additin : First Principles__

→ __Vector Addition : Component Form__

→ __Triangular Law__

→ __Parallelogram Law__

• Multiplication of Vector by Scalar

→ __Scalar Multiplication__

→ __Standard Unit Vectors__

→ __Vector as Sum of Vectors__

→ __Vector Component Form__

• Vector Dot Product

→ __Introduction to Vector Multiplication __

→ __Cause-Effect-Relation__

→ __Dot Product : First Principles__

→ __ Dot Product : Projection Form__

→ __ Dot Product : Component Form__

→ __Dot Product With Direction__

• Vector Cross Product

→ __Vector Multiplication : Cross Product __

→ __Cross Product : First Principles__

→ __Cross Product : Area of Parallelogram__

→ __Cross Product : Component Form__

→ __Cross Product : Direction Removed__