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 vectors
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 = x_{1}+ x_{2}
y = y_{1}+ y_{2}
z = z_{1}+ z_{2}
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 = x_{1}  x_{2}
y = y_{1}  y_{2}
z = z_{1 } z_{2}
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 xy) directed in accordance with RightHand Rule: when right palm is halfbent 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
