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:  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 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: 