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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:  ->{a}

The two most common ways to denote vectors are as follows:

Representing a vector using Polar coordinates (length and angle)



 ->{a}=(L,_|theta)

where L is the length of the vector and _|theta is the angle of the vector in respect to the x-axis.

Example:
Vector with direction and magnitude ->{a}=(5,_|37^o)

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:

 ->{v}=(x,y)

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.

Eksempel:
Vector with cartesian coordinates ->{v}=(4,3)

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:

 ->{a}=(L,_|v)=(L*cos(v),L*sin(v))

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.

 ->{AB}=(x_b-x_a,y_b-y_a)