Vector Algebra : Standard Unit Vectors

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Standard Unit Vectors

what you'll learn...

Overview

Standard Unit Vectors

» $3$ $3$ orthogonal axes are represented with $3$ $3$ unit vectors
    → x-axis $i$ $i$
    → y-axis $j$ $j$
    → z-axis $k$ $k$

unit means "one"

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

• $3$ $3$ m north

• $4.2$ $4.2$ m south

• $\frac{3}{4}$ $3/4$ m north

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

The answer is ' $5$ $5$ m '– as the person moved $5$ $5$ 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 $90^{\circ}$ $90^@$ 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 $3$ $3$ orthogonal components in 3D coordinate space and $3$ $3$ orthogonal axes are defined for that.

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

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

summary

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

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