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 = 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