Hamza

March 13, 2021

Chapters

Basic Formulas
Magnitude of a Vector
Angle Between Two Vectors
Orthogonal Vectors
Area of a Parallelogram
Area of a Triangle

Basic Formulas

Components

$\vec { AB } = ({ x }_{ 2 } - { x }_{ 1 }, { y }_{ 2 } - { y }_{ 1 }, { z }_{ 2 } - { z }_{ 1 })$

Magnitude or Length

$\vec { u } = ({ u }_{ 1 }, { u }_{ 2 }, { u }_{ 3 })$

$\left | \vec { u } \right |= \sqrt { { u }_{ 1 }^{ 2 } + { u }_{ 2 }^{ 2 } + { u }_{ 3 }^{ 2 } }$

Distance between two points

$d(A, B ) = \left | \vec { AB } \right | = \sqrt { { ({ x }_{ 2 } - { x }_{ 1 }) }^{ 2 } + { ({ y }_{ 2 } - { y }_{ 1 }) }^{ 2 } + { ({ z }_{ 2 } - { z }_{ 1 }) }^{ 2 } }$

Unit Vector

${ \vec { u } }_{ \vec { v } } = \frac { \vec { v } }{ \left | \vec { v } \right | }$

Vector Addition

$\vec { u } = ({ u }_{ 1 }, { u }_{ 2 }, { u }_{ 3 }) \qquad \vec { v } = ({ v }_{ 1 }, { v }_{ 2 }, { v }_{ 3 })$

$\vec { u } + \vec { v } = ({ u }_{ 1 } + { v }_{ 1 }, { u }_{ 2 } + { v }_{ 2 }, { u }_{ 3 } + { v }_{ 3 })$

Scalar Multiplication

$k . \vec { u } = (k { u }_{ 1 } + k { u }_{ 2 } + k { u }_{ 3 })$

Linearly Dependent Vectors

$\vec { u } = k \vec { v } \qquad ({ u }_{ 1 }, { u }_{ 2 }, { u }_{ 3 }) = (k { v }_{ 1 } + k { v }_{ 2 } + k { v }_{ 3 })$

$\frac { { u }_{ 1 } }{ { v }_{ 1 } } = \frac { { u }_{ 2 } }{ { v }_{ 2 } } = \frac { { u }_{ 3 } }{ { v }_{ 3 } } = k$

$\begin{vmatrix} { u }_{ 1 } & { u }_{ 2 } & { u }_{ 3 } \\ { v }_{ 1 } & { v }_{ 2 } & { v }_{ 3 } \\ { w }_{ 1 } & { w }_{ 2 } & { w }_{ 3 } \end{vmatrix} = 0$

Linearly Independent Vectors

$\begin{vmatrix} { u }_{ 1 } & { u }_{ 2 } & { u }_{ 3 } \\ { v }_{ 1 } & { v }_{ 2 } & { v }_{ 3 } \\ { w }_{ 1 } & { w }_{ 2 } & { w }_{ 3 } \end{vmatrix} \neq 0$

Dot Product

$\vec { u } . \vec { v } = \left | \vec { u } \right | . \left | \vec { v } \right | . \cos { \alpha }$

$\vec { u } . \vec { v } = { u }_{ 1 } . { v }_{ 1 } + { u }_{ 2 } . { v }_{ 2 } + { u }_{ 3 } . { v }_{ 3 }$

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Magnitude of a Vector

$\left | \vec { u } \right | = \sqrt { \vec { u } . \vec { u } } = \sqrt { { u }_{ 1 } . { u }_{ 1 } + { u }_{ 2 } . { u }_{ 2 } + { u }_{ 3 } . { u }_{ 3 } } = \sqrt { { u }_{ 1 }^{ 2 } + { u }_{ 2 }^{ 2 } + { u }_{ 3 }^{ 2 } }$

Angle Between Two Vectors

$\cos { \alpha } = \frac { { u }_{ 1 } . { v }_{ 1 } + { u }_{ 2 } . { v }_{ 2 } + { u }_{ 3 } . { v }_{ 3 } }{ \sqrt { { u }_{ 1 }^{ 2 } + { u }_{ 2 }^{ 2 } + { u }_{ 3 }^{ 2 } } . \sqrt { { v }_{ 1 }^{ 2 } + { v }_{ 2 }^{ 2 } + { v }_{ 3 }^{ 2 } } }$

Orthogonal Vectors

$\vec { u } . \vec { v } = ({ u }_{ 1 } . { v }_{ 1 } + { u }_{ 2 } . { v }_{ 2 } + { u }_{ 3 } . { v }_{ 3 }) \times 0 = 0$

Direction Cosine

$\cos { \alpha } = \frac { x }{ \sqrt { { x }^{ 2 } + { y }^{ 2 } + { z }^{ 2 } } }$

$\cos { \beta } = \frac { y }{ \sqrt { { x }^{ 2 } + { y }^{ 2 } + { z }^{ 2 } } }$

$\cos { \gamma } = \frac { z }{ \sqrt { { x }^{ 2 } + { y }^{ 2 } + { z }^{ 2 } } }$

$\cos^{ 2 }{ \alpha } + \cos^{ 2 }{ \beta } + \cos^{ 2 }{ \gamma } = 1$

Cross Product

$\vec { u } \times \vec { v } = \begin{vmatrix} \vec { i } & \vec { j } & \vec { k } \\ { u }_{ 1 } & { u }_{ 2 } & { u }_{ 3 } \\ { v }_{ 1 } & { v }_{ 2 } & { v }_{ 3 } \end{vmatrix} = \begin{vmatrix} { u }_{ 2 } & { u }_{ 3 } \\ { v }_{ 2 } & { v }_{ 3 } \end{vmatrix} \vec { i } - \begin{vmatrix} { u }_{ 1 } & { u }_{ 3 } \\ { v }_{ 1 } & { v }_{ 3 } \end{vmatrix} \vec { j } + \begin{vmatrix} { u }_{ 1 } & { u }_{ 2 } \\ { v }_{ 1 } & { v }_{ 2 } \end{vmatrix} \vec { k }$

$\left | \vec { u } \times \vec { v } \right | = \left | \vec { u } \right | \left | \vec { v } \right | \sin { \alpha }$

Area of a Parallelogram

$A = \left | \vec { u } \right | . h = \left | \vec { u } \right | \left | \vec { v } \right | \sin { \alpha } = \left | \vec { u } \times \vec { v } \right |$

Area of a Triangle

$A = \frac { 1 }{ 2 } \left | \vec { u } \times \vec { v } \right |$

Scalar Triple Product

$[\vec { u }, \vec { v }, \vec { w }] = \vec { u } . (\vec { v } \times \vec { w })$

$[\vec { u }, \vec { v }, \vec { w }] = \begin{vmatrix} { u }_{ 1 } & { u }_{ 2 } & { u }_{ 3 } \\ { v }_{ 1 } & { v }_{ 2 } & { v }_{ 3 } \\ { w }_{ 1 } & { w }_{ 2 } & { w }_{ 3 } \end{vmatrix}$

Volume of a Parallelepiped

$\vec { u } = (3, -2, 5) \quad \vec { v } = (2, 2, -1) \quad \vec { w } = (-4, 3, 2)$

Volume of a tetrahedron

$V = \frac { 1 }{ 6 } \begin{vmatrix} { u }_{ 1 } & { u }_{ 2 } & { u }_{ 3 } \\ { v }_{ 1 } & { v }_{ 2 } & { v }_{ 3 } \\ { w }_{ 1 } & { w }_{ 2 } & { w }_{ 3 } \end{vmatrix}$

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.

Comments

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Sathvik

April

Displacement

Unknown

May

Thanks, it is very helpful. A few more formulas need to be added though.

Anonymous

June

Thank You!

Sai

September

I need scaler triple product breifl6

I need scaler triple product breifly

Swarnaprava Nayak

October

I want Vectors full chapter briefly

Vector Formulas

Basic Formulas

Components

Magnitude or Length