# Formulas for Vectors

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.

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:

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

More on Vector Addition.

• Subtraction The subtraction of vectors and is defined by

More on Vector Subtraction

More on Adding and Subtracting vectors.

• 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: