Direction Vectors

Introduction:

In mathematics a direction vector that describes a line segment D is any vector of the direction vector,

AB?

Where, A and B are two distinct points on the line D. If v is a direction vector for the D, so is kv for any nonzero scalar k; and these are in fact all of the direction vectors for the line D. Under the some definitions, the direction vector is required to be a unit vector, in which case each line has exactly two direction vectors, which are negatives of each other equal in magnitude, opposite in direction. (Source:Wikipedia)

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Direction vector for a line in two-dimensional:


Any other line in two-dimensional Euclidean space can be described as the set of solutions to an equation of the form

ax + by + c = 0

Where a, b, c are real numbers. Then one direction vector of (D) is (- b,a). Any multiple of (- b, a) is also a direction vector.


Basic properties:


The following section uses the Cartesian coordinate system with basis vectors

a=a1e1+a2e2+a3e3 and
b=b1e1+b2e2+b3e3
Are equal if

a1=b1, a2=b2, a3=b3.


Magnitude and Direction of a Vector:


Let v can be a vectors given in component form by

                        v = < a,b >

The magnitude of the || v || of vector v is given by

                 || v || = sort (a 2 + b 2)

and the direction of the vector v is angle t in standard position and in counterclockwise direction is such that

                  tan(t) = v / u

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Vector methods:


Vectors and vector addition
Unit vectors
Base vectors and vector components
Rectangular components in 2-D
Rectangular coordinates in 3-D
Direction cosines
A vector connecting two points
Dot product
Rectangular coordinates
Projection of a vector onto a line
The cross product
The triple product
Rectangular coordinates
Triple vector product

Example:

The equation of a line is 2x - 3y + 15 = 0.

2x-3y=-15

2(-3)-3(3)=-15

-6-9=-15

So (-3, 3) is direction vectors for this line