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Properties of Magnitude of a Vector


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Overview

Properties of Magnitude of Vectors

 »  Magnitude of null vectors |0|=0

 »  Magnitude of a negative |-p|=|p|

 »  Magnitude of an unit vector |p^|=1

properties

Null vector is 0=0i+0j+0k. The magnitude of null vector is |0|=02+02+02=0.

Magnitude of a Null vector: Null vector is 0=0i+0j+0k. The magnitude |0|=0

The magnitude of a null vector is zero.


Given a vector p, The magnitude of negative of the vector =|p|

p=pxi+pyj+pzk

The negative of p is -p=-pxi-pyj-pzk and so
|-p|
    =(-px)2+(-py)2+(-pz)2
    =(px)2+(py)2+(pz)2
    =|p|

Magnitude of negative vector: For a vector p, the magnitude of the negative |-p|=|p|

Magnitude of the negative of a vector equals magnitude of the vector.


The magnitude of a proper vector 0

Given a proper vector p=pxi+pyj+pzk,

By definition, a proper vector is not a null vector. That is, at least one of the components px,py,pz is non-zero, and can be a positive or negative value. So the magnitude (px)2+(py)2+(pz)2 is not zero and positive.

Magnitude of Proper Vector: For a proper vector p, the magnitude |p|0 and |p|>0

Magnitude of a proper vector is not zero and positive.


The magnitude of a unit vector is 1

By definition, magnitude of an unit vector is 1.

Magnitude of a unit vector: For a unit vector p, |p|=1

Magnitude of an unit vector is 1

summary

 »  Magnitude of null vectors |0|=0

 »  Magnitude of a negative |-p|=|p|

 »  Magnitude of an unit vector |p^|=1

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