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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 )

Vector and its components

The notation of vector with coordinates has the form


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

|| =

Adding vectors

Adding of vectors  and  is sum vector


The sum vector is determined by Parallelogram Rule of addition

Parallelogram Rule of addition

Magnitude of sum vector is defined by Law of Cosines


 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

Polygon Rule of addition

Subtracting vectors

Subtracting of vectors  and   is vector difference


The vector difference is determined by Triangle Method of subtraction

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:


2) If  then

3) If  then


Vector product

Vector (or cross) product of vectors  and  is vector


Vector product

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:


2) If  then

3) If  then magnitude of vector product is