Vectors in a plane
While scalars in physics and mathematics are one-dimensional quantities described with only one real number, a vector is described by two or three numbers, depending on if it is a two- or three-dimensional vector.
The two numbers of a two-dimensional vector characterize respetively the direction and the magnitude (the length) of the vector.
Vectors are often used in physics to describe forces or acceleration.
A vector is an arrow in a coordinate system having a given direction and a given length. It has no starting point. The only thing that matters is its direction and magnitude.
Denotation of vectors
A vector is usually written with an arrow above a letter:
The two most common ways to denote vectors are as follows:
Representing a vector using Polar coordinates (length and angle)
where L is the length of the vector and is the angle of the vector in respect to the x-axis.
Representing a vector using Cartesian coordinates (x and y)
The most common and practical way to denote a vector is with Cartesian coordinates. They work the same way as when you indicate the slope of a line, where the change of x and y is specified:
x indicates how much the vector will change in the x-axis direction and y indicates how much the vector will change in the y-axis direction.
Converting between the two denotations of vectors
It is possible to convert between Polar coordinates and Cartesian coordinates of a vector. The formula is given by:
The Cartesian coordinates are used in most formulas.
Defining a vector with two points given
It is possible to define a vector, if two points, A(xa,ya) and B(xb,yb), are given in a coordinate system.