Some of the most important formulas for vectors such as the magnitude, the direction, the unit vector, addition, subtraction, scalar multiplication and cross product are presented.
Vector Defined by two Points
The components of a vector defined by two points and are given as follows:
In what follows , and are 3-D vectors given by their components as follows
Magnitude of a Vector
The magnitude of vector written as is given by
Unit Vector
A unit vector is a vector whose magnitude is equal to 1.
The unit vector that has the same direction a vector is given by
Direction of a Vector
In 3-D, the direction of a vector is defined by 3 angles α , β and γ (see Fig 1. below) called direction cosines.
Fig1. - Direction cosine of a vector.
These are the angles between the vector and the positive x-, y- and z- axes respectively of a rectangular system. The cosine of these angles, for vector ,are given by:
Fig1. - Angle of a 2-D vector.
In 2-D, the direction of a vector is defined as an angle that a vector makes with the positive x-axis. Vector (see Fig 2. on the right) is given by
taking into account the signs of Ax and Ay to determine the quadrant where the vector is located.
Operations on Vectors
Addition
The addition of vectors and is defined by
Multiply Vector by a Scalar
The multiplication of vectors by a scalar k is defined by
Scalar Product of Vectors
Definition
The Scalar (or dot) product of two vectors and is given by
where θ is the angle between vectors A and B
Given the coordinates of vectors and , it can be shown that
Properties of Scalar Product
Orthogonal Vectors
Two vectors and are orthogonal if and only if
Angle Between Two Vectors
If θ is the angle made by two vectors and , then
Cross Product
The cross product of two vectors and is a vector orthogonal to both vectors and is given by
Properties of Cross Product
The cross product is a vector and there may a need as in eletromagnetism and many other topics in physics to find the orientation of this vector. Use the right hand rule to find the orientation of the cross product: point the index in the direction of vector A, the middle finger in the direction of vector B and the direction of the cross product A × B is in the same direction of the thumb.
Geometrical Meaning of Cross Product
The area of a parallelogram defined by vectors and is the magnitude of their cross product given by:
defined by two points 
and 
are given as follows:
, 
and 
are 3-D vectors given by their components as follows
is given by
that has the same direction a vector 
(see Fig 2. on the right) is given by
