Formulas for Vectors

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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.

cosine directions of a 3d vector
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:

angle of a 2-D vector
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.
right hand for cross product
Geometrical Meaning of Cross Product The area of a parallelogram defined by vectors and is the magnitude of their cross product given by:


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Updated: 26 September 2015

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