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# Vectors

Vector is a quantity specified by magnitude plus direction in space

## Properties of vectors

Any vector  is uniquely specified by its three components x, y, z which are projections of the vector on coordinate axes with unit vectors  (such that )

The notation of vector with coordinates has the form

is

Magnitude of vector  (or its length) is defined by Pythagorean Theorem

|| =

Adding of vectors  and  is sum vector

The sum vector is determined by Parallelogram Rule of addition

Magnitude of sum vector is defined by Law of Cosines

where:

and  are magnitudes of the vectors  and

is angle between them

Components of sum vector:

x = x1+ x2

y = y1+ y2

z = z1+ z2

The parallelogram rule of addition is partial case of general Polygon Rule used for adding several vectors

## Subtracting vectors

Subtracting of vectors  and   is vector difference

The vector difference is determined by Triangle Method of subtraction

Magnitude of vector difference

Components of vector difference:

x = x1 - x2

y = y1 - y2

z = z1 - z2

## Scalar Product

Scalar (or dot) product of vectors  and  is scalar quantity defined by

where  is angle between vectors  and

Properties of scalar product:

1)

2) If  then

3) If  then

4)

## Vector product

Vector (or cross) product of vectors  and  is vector

The vector  is normal to plane in which the vectors  and lie (plane x-y) directed in accordance with Right-Hand Rule: when right palm is half-bent from  to  its thumb shows the direction of vector

Magnitude of vector

where  is angle between vectors  and

Components of vector product:

Properties of vector product:

1)

2) If  then

3) If  then magnitude of vector product is