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Vector

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You needs a lot of things to express anything in physics. Physics world is all about expressing the facts behind the qualitative and quantitative models. Quantitative facts can be expressed using physical quantities like speed displacement, temperature, length etc.
Vector
There are basically two types of physical quantities - scalars and vectors. vectors tells about magnitude and direction of any physical quantity. The resultant is that what we get by combining the vector components. They are many vector quantities like displacement, velocity, acceleration, momentum, force etc. Lets study more about this.

Unit Vector

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It is a type of vector having magnitude as one. It is expressed with hat given as $\hat{a}$ where $\hat{a}$ = 1. It always indicates a specific direction.
It is given as
Unit Vector FormulaHere a is the given vector, |a| is the magnitude of vector a and $\hat{a}$ is the unit vector.

Zero Vector

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Zero vector called as null vector is that vector having magnitude as zero in any arbitrary direction. It is represented as $\vec{0}$. Its a dot with zero displacement. Fig below represents it
Zero Vector

Parallel Vector

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Parallel vector word itself tells that the vectors are parallel. Two vectors are parallel if they have same or opposite direction. Their cross product is always zero. Figure below illustrates it :
Parallel Vectors
If $\vec{a}$ and $\vec{b}$ are the parallel vectors then

Vector addition

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The vector addition is a mathematical way of adding two vectors to get the resultant. They can only be added if they are directed in same direction. Vector addition can be using triangle law of vector addition and parallelogram of vector addition.
Vector Addition
If $\vec{a}$ and $\vec{b}$ are the vectors given then the vector addition will be the magnitude of two given vectors. The vector addition is
Vector Addition Formula

Vector Subtraction

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Vector subtraction is operating the subtraction operation for the given vectors. This operation is for the vectors directed in opposite direction.
Vector Subtraction
If $\vec{a}$ and $\vec{b}$ are given vectors then Vector subtraction is
Vector Subtraction Formula

Vector Multiplication

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Vector multiplication is multiplying the vectors that changes in magnitude but it does not alter its direction.
Vector Multiplication
There are two types of vector multiplication
  1. Cross product
  2. Dot product.

Vector Dot Product

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Vector Dot product is the value to express the angular relationship between the given two vectors. It is a inner product that multiply vectors together with the resultant being scalar.
Vector Dot Product
The dot product for two vectors $\vec{a}$ and $\vec{b}$ is
Vector Dot Product FormulaHere $\theta$ is the angle between the given two vectors $\vec{a}$ and $\vec{b}$.

Vector Cross Product

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Vector cross product also called as cross product is the vector product of operation of two vectors.
Vector Cross Product
If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ then cross product is
Vector Cross Product Formula