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Welcome to our comprehensive guide on vector algebra formulas! Whether you’re a student, educator, or just curious about vectors, this PDF compilation contains 100% free essential formulas to help you master vector concepts.

Physical Quantity: Property of a material or system that we can measure using numbers. These measurements have two important components:

Magnitude: This represents the size or amount of the property.
Unit: The unit gives meaning to the numerical value and allows us to compare and communicate data effectively.

There are mainly two type of physical quantity:

Scalar Quantity: completely defined by only Magnitude.
Vector Quantity: defined by Magnitude and Direction.

Vectors Representation/Notation

Vector Quantity has both Magnitude and Direction and must be follow vector algebra.

$\vec{A} = A\hat{A}$

where A = magnitude of $\vec{A}$ and $\hat{A}$ = direction (unit vector)

Comparison between Scalar and Vector Quantity

Scalar Quantity	Vector Quantity
Defined by only Magnitude	Magnitude + Direction
Follow Normal Algebraic Rules	Follow Vector Algebraic Rules
Scalar quantities are added, subtracted, or divided by Algebraically	Vector quantities are added and subtracted Geometrically
Examples: Length, Distance, Speed, Mass, Volume, Time, Time, Temperature, Work, Energy, Power, Electric Current, Charge, Potential, Flux, etc.	Examples: Position, Displacement, Velocity, Weight, Acceleration, Electric Field, Surface Area, Force, Torque, etc.

comparison between Scalar and Vector Quantity

Physical Quantities having different values in different direction known as Tensor.

Examples: Moment of inertia, Refractive Index, Density Stress and Stain, etc.
Key Point:

Types of Vectors

Sr.No.	Vector Type	Definition
1.	Equal Vectors $\vec{A}=\vec{B}$	Magnitudes $A=B$ & Direction $\hat{A}=\hat{B}$
2.	Opposite Vectors $\vec{A}=-\vec{B}$	Two vectors $\vec{A}$ & $\vec{B}$ are said to be opposite vectors if thier magnitudes are same but direction is opposite.
3.	Position Vector	$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$
4.	Displacement Vector	$\vec{d}=\vec{r_2}-\vec{r_1}$
5.	Unit Vector	$\hat{A}=\frac{\vec{A}}{A}$ Magnitude: Unit (1), Direction:
6.	Orthogonal Unit Vectors	The unit vector along x-axis $(\hat{i})$ , y-axis $(\hat{j})$ & z-axis $(\hat{k})$ called orthogonal vectors. and $\hat{i}\perp\hat{j}\perp\hat{k}$
7.	Zero / Null Vector ( $\vec{0}$ )	Magnitude: Zero (0) Direction: Arbitrary
8.	Collinear Vectors	Two Parallel or Antiparallel vectors are called colinear vectors. $\vec{A}=\pm\lambda\vec{B}$
9.	Coplanar Vectors	Vectors having same Plane or Parallel Plane are called coplanar. If $\vec{A}$ , $\vec{B}$ and $\vec{C}$ are coplanar then, $V=\vec{A}.[\vec{B}\times\vec{C}]=0$
10.	Polar Vector	These vectors have defined initial point. Examples: displacement, force, position, velocity, linear momentum..etc.
11.	Axial Vector	These vectors always along an axis. Examples: angular momentum, angular velocity, torque, etc.

Addition of Vectors

Triangle Law of vector Addition: If two vectors A and B are represented as two sides of a triangle taken in order, then the third side of the triangle represents the resultant vector R: $\vec{R}=\vec{A}+\vec{B}$
Parallelogram law of vector addition: If two vectors A and B represented as adjacent sides of a parallelogram, then the diagonal of the parallelogram represents the resultant vector R: $\vec{R}=\vec{A}+\vec{B}$

Magnitude of the Resultant Vector: $R=\vec{R}= |\vec{A}+\vec{B}|=\sqrt{A^2 + B^2+2ABcos\theta}$

Direction of Resultant Vector: $tan\alpha=\frac{Bsin\theta}{A+Bcos\theta}$

Case (1): If $\vec{A}$ & $\vec{B}$ are parallel i.e. $\theta =0°$ then,
$|\vec{R}|= A+B$ (maximum resultant)

Case (2): If $\vec{A}$ & $\vec{B}$ are Perpendicular i.e. $\theta =90°$ then,
$|\vec{R}|=\sqrt{ A^2+B^2}$

Case (3): If $\vec{A}$ & $\vec{B}$ are Anti-parallel i.e. $\theta =180°$ then,
$R=A\sim B$ (minimum resultant)

Special Cases to related vector addition

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Subtraction of vectors

Magnitude of the Resultant Vector: $R=\vec{R}= |\vec{A}-\vec{B}|=\sqrt{A^2 + B^2-2ABcos\theta}$

Direction of Resultant Vector: $tan\alpha_2=\frac{Bsin\theta}{A-Bcos\theta}$

Case (1): If $\vec{A}$ & $\vec{B}$ are parallel i.e. $\theta =0°$ then,
$|\vec{R}|= A\sim B$ (minimum resultant)

Case (2): If $\vec{A}$ & $\vec{B}$ are Perpendicular i.e. $\theta =90°$ then,
$|\vec{R}|=\sqrt{ A^2+B^2}$

Case (3): If $\vec{A}$ & $\vec{B}$ are Anti-parallel i.e. $\theta =180°$ then,
$R=A+ B$ (maximum resultant)

Special Cases to related vector subtraction

Formulas of Scalar Product of Two Vectors

If $\vec{A}$ and $\vec{B}$ having angle $\theta$ , then their scalar product is defined as:

$\vec{A}.\vec{B}=ABcos\theta$

$\theta=cos^{-1}[\frac{\vec{A}.\vec{B}}{AB}]$

In terms of components, if $\vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}$ and $\vec{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k}$

$\vec{A}.\vec{B}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}.\vec{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k}=[A_xB_x+A_yB_y+A_zB_z]$

Case (1): If $\vec{A}$ & $\vec{B}$ are parallel i.e. $\theta =0°$ then,
$\vec{A}.\vec{B})_max= A.B$

Case (2): If $\vec{A}$ & $\vec{B}$ are Perpendicular (orthogonal) i.e. $\theta =90°$ then,
$\vec{A}.\vec{B})_min= 0$

Special Cases to related dot product of two vector

Scalar Product Properties

It is always a scalar.
If $\theta <90^o$ (acute) then dot product always be positive.
If $90^o<\theta <180^o$ (obtuse) then dot product always be negative.
$\vec{A}.\vec{B}=\vec{B}.\vec{A}$ i.e. it is commutative.
$\vec{A}.(\vec{B}+\vec{C})=\vec{A}.\vec{B}+\vec{A}.\vec{C}$ i.e. it is distributive.
The scalar product of a vector by itself is termed as self dot product and is given by, $(\vec{A})^2=\vec{A}.\vec{B}=AAcos\theta=A^2$ i.e. $A=\sqrt{\vec{A}.\vec{A}}$
The dot product of same orthogonal unit vectors : $\hat{i}.\hat{i}=\hat{j}.\hat{j}=\hat{k}.\hat{k}=1$
The dot product of orthogonal unit vectors : $\hat{i}.\hat{j}=\hat{j}.\hat{k}=\hat{k}.\hat{i}=0$

Examples of Scalar Product

Work : $w =\vec{F}.\vec{d}=Fdcos\theta$
Energy : $U_E=-\vec{p}.\vec{E}$ & $U_B=-\vec{M}.\vec{B}$
Power : $P=\vec{F}.\vec{v}$
Magnetic Flux : $d\Phi=\vec{B}.d\vec{s}$

Formulas of Vector Product of Two Vectors

$\vec{A}\times\vec{B}=ABsin\theta\hat{n}$

$\theta=cos^{-1}[\frac{\vec{A}\times\vec{B}}{AB}]$

In terms of components: if $\vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}$ and $\vec{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k}$

$\vec{A}\times\vec{B}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\A_x & A_y & A_z\\B_x & B_y & B_z\end{vmatrix}$

$= \hat{i}(A_y B_z - A_z B_y)+\hat{j}(A_z B_x - A_x B_z)+\hat{k}(A_x B_y - A_y B_x)$

Case (1): If $\vec{A}$ & $\vec{B}$ are parallel i.e. $\theta =0°$ or antiparallel i.e. $\theta=180°$ then,
$[\vec{A}\times\vec{B}]_min= A0°$

Case (2): If $\vec{A}$ & $\vec{B}$ are Perpendicular (orthogonal) i.e. $\theta =90°$ then,
$\vec{A}\times\vec{B})_max=AB\hat{n}$

Special Cases to related dot product of two vector

Vector Product Properties

It is always a vector.
$\vec{A}\times\vec{B}=-\vec{B}\times\vec{A}$ i.e. Vector product of two vectors is not commutative.
$\vec{A}\times(\vec{B}+\vec{C}=\vec{A}\times\vec{B}+\vec{A}\times\vec{C}$ i.e. The vector product is distributive when the order of the vectors is strictly maintained.
The self cross product, i.e., product of a vector by itself vanishes, i.e., is null vector $\vec{A}\times\vec{A}=AAsin0°\hat{n}=\vec{0}$
The dot product of same orthogonal unit vectors : $\hat{i}\times\hat{i}=\hat{j}\times\hat{j}=\hat{k}\times\hat{k}=0$
The dot product of orthogonal unit vectors : $\hat{i}\times\hat{j}=\hat{k}, \hat{j}\times\hat{k}=\hat{i}, \hat{k}\times\hat{i}=\hat{j}$

Examples of Cross products

$\vec{\tau}=\vec{r}\times\vec{F}$
$\vec{v} =\vec{\omega}\times\vec{r}$
$\vec{L}=\vec{r}\times\vec{p}$

Comparison between Scalar and Vector products

All Essential Vector Algebra Formulas PDF [100% Free]

Vectors Representation/Notation

Comparison between Scalar and Vector Quantity