Let A = (1,3), B = (6,6)
___
Let v = AB
||v|| represents the magnitude of v
which is the length of the line drawn.
so 
Let C = (4,-1) and D = (9,2)
__
Let u = CD
Then 
Since u and v travel the same direction
and have the same magnitude, they are equivalent vectors.
A vector with its initial point at the vertex is in standard
position.
The angle between the vector and the positive x-axis
is the direction angle.
To actually know if u and v are
travelling in the same direction, we should find their
direction
angles.
B
A --------------------------O
Every vector has a vertical and horizontal component. OA
is
the horizontal component. OB is
___
the vertical component. A is the direction angle for
vector AB.
Thus, using trigonometric functions, OA = ||AB|| cos A
and OB = ||AB|| sin A.
The component form of a vector is written <a,b>.
The sum or resultant of two vectors is found using parallelogram addition or triangular addition.
Triangular addition is done by placing the second first so that its initial point touches the terminal point of the first. Then draw a vector from the initial point of the first vector to the terminal point of the second vector.
Parallelogram addition is done by placing both vectors in standard position and drawing the parallelogram formed by using these as two sides. The resultant is the diagonal of the parallelogram that is drawn from the origin. This procedure has some advantages over triangular addition when other parts of the triangle need to be found.
A unit vector is a vector with length 1. It can
be found by dividing a vector by its magnitude.
If u is a unit vector in standard position,
then u = < cos A, sin A >.
i = < 1,0 > and j = < 0,1 > are standard unit vectors.
Vector arithmetic:
Let u = < a,b > and v = < c,d >
u + v = < a,b > + < c,d > = < a
+
c, b + d >
u - v = < a,b > - < c,d >
=
< a - c, b - d >
ku = k< a,b > = < ka, kb
>