Vector Algebra : Properties of Magnitude of a Vector

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Properties of Magnitude of a Vector

what you'll learn...

Overview

Properties of Magnitude of Vectors

» Magnitude of null vectors $| \vec{0} |$ $|vec(0)|$ $= 0$ $= 0$

» Magnitude of a negative $| - \vec{p} | = | \vec{p} |$ $|-vec(p)| = |vec(p)|$

» Magnitude of an unit vector $| \hat{p} | = 1$ $|hat p|= 1$

properties

Null vector is $\vec{0} = 0 i + 0 j + 0 k$ $vec 0 = 0i+0j+0k$ . The magnitude of null vector is $| \vec{0} | = \sqrt{0^{2} + 0^{2} + 0^{2}} = 0$ $|vec 0| = sqrt(0^2+0^2+0^2) = 0$ .

Magnitude of a Null vector: Null vector is $\vec{0} = 0 i + 0 j + 0 k$ $vec 0 = 0i+0j+0k$ . The magnitude $| \vec{0} | = 0$ $|vec 0| = 0$

The magnitude of a null vector is zero.

Given a vector $\vec{p}$ $vec p$ , The magnitude of negative of the vector $= | \vec{p} |$ $=|vec p|$

$\vec{p} = p_{x} i + p_{y} j + p_{z} k$ $vec p = p_x i + p_y j + p_z k$

The negative of $\vec{p}$ $vec p$ is $- \vec{p} = - p_{x} i - p_{y} j - p_{z} k$ $-vec p = -p_x i - p_y j - p_z k$ and so
$| - \vec{p} |$ $|-vec p|$
$= \sqrt{{(- p_{x})}^{2} + {(- p_{y})}^{2} + {(- p_{z})}^{2}}$ $quad quad = sqrt((-p_x)^2+(-p_y)^2+(-p_z)^2)$
$= \sqrt{{(p_{x})}^{2} + {(p_{y})}^{2} + {(p_{z})}^{2}}$ $quad quad = sqrt((p_x)^2+(p_y)^2+(p_z)^2)$
$= | \vec{p} |$ $quad quad = |vec p|$

Magnitude of negative vector: For a vector $\vec{p}$ $vec p$ , the magnitude of the negative $| - \vec{p} | = | \vec{p} |$ $|-vec p| = |vec p|$

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

The magnitude of a proper vector $\neq 0$ $!=0$

Given a proper vector $\vec{p} = p_{x} i + p_{y} j + p_{z} k$ $vec p = p_x i + p_y j + p_z k$ ,

By definition, a proper vector is not a null vector. That is, at least one of the components $p_{x}, p_{y}, p_{z}$ $p_x, p_y, p_z$ is non-zero, and can be a positive or negative value. So the magnitude $\sqrt{{(p_{x})}^{2} + {(p_{y})}^{2} + {(p_{z})}^{2}}$ $sqrt((p_x)^2+(p_y)^2+(p_z)^2)$ is not zero and positive.

Magnitude of Proper Vector: For a proper vector $\vec{p}$ $vec p$ , the magnitude $| \vec{p} | \neq 0$ $|vec p| != 0$ and $| \vec{p} | > 0$ $|vec p| > 0$

Magnitude of a proper vector is not zero and positive.

The magnitude of a unit vector is $1$ $1$

By definition, magnitude of an unit vector is 1.

Magnitude of a unit vector: For a unit vector $\vec{p}$ $vec p$ , $| \vec{p} | = 1$ $|vec p| =1$

Magnitude of an unit vector is 1

summary

» Magnitude of null vectors $| \vec{0} |$ $|vec(0)|$ $= 0$ $= 0$

» Magnitude of a negative $| - \vec{p} | = | \vec{p} |$ $|-vec(p)| = |vec(p)|$

» Magnitude of an unit vector $| \hat{p} | = 1$ $|hat p|= 1$

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