Vectors in a plane
Definition of a vector in a plane
A
vector in a plane is a
directed straight line segment in the plane.
Any straight line segment in a plane is defined by the two its endpoints. A straight line segment in a plane become the
directed straight line segment when one of its two endpoints is declared as the
initial point while the other endpoint is declared as the
terminal point.
Vectors are named by their initial point and terminal point usually. In naming vectors by their initial point and terminal point the initial point goes first.
Thus the
vector AB in a plane is the directed straight line segment with the initial point
A and the terminal point
B (see
Figure 1).
Each ordered pair of points (
A,
B) in a plane defines the vector
AB which is the directed segment
AB with the initial point
A and the terminal point
B.
A vector is drawn as a line segment with an arrow at the terminal point as shown in the
Figure 1.
Sometimes vectors are named by a single lovercase letter like
a instead of
AB.
Each vector in a plane has the
length (or the
magnitude) and the
direction. The length of the vector is the length of its straight line segment. It is a nonnegative real number. The length of a vector
a is usually denoted as 
a.
Figure 1. The vector in a plane

Figure 2. Two equal vectors AB and CD in a plane

Figure 3. Two opposite vectors AB and CD in a plane

Two vectors
AB and
CD in a plane are called
equal (or
equivalent),
AB=
CD, if the segments
AB and
CD are parallel and codirected, and have equal length (see
Figure 2).
Two vectors in a plane are called
opposite if the corresponding segments are parallel, have opposite directions, and have equal length (see
Figure 3).
It is clear that if the vector
AB is equal to the vector
CD and the vector
CD is equal to the vector
EF then the vector
AB is equal to the vector
EF.
A vector whose initial and terminal points coincide is called the
zero vector. The standard designation for the zero vector is
0.
The length of the zero vector is zero: 
0 = 0. The direction of the zero vector is assumed to be arbitrary.
From the point of view of this definition of vectors equality, the concrete location of the vector's initial and terminal points does not matter. For example, two directed straight line segments in the
Figure 2 represent the same vector. Moreover, there are infinitely many straight segments in a plane having the same length and direction. All of them are distinguished by the location of the initial point only, and all of them represent the same unique vector. If you want, you may think that one of these vectors, which has the origin of the coordinate system as the initial point, represents all of this infinite set of equivalent vectors. Only the direction and the magnitude do matter in distinguishing vectors.
For the quantitative definitions of the vector length and direction in a coordinate plane see the lesson Vectors in a coordinate plane under the current topic in this site.
Definition of adding vectors in a plane
If the endpoint of the vector
AB is the initial point of the vector
BC then the sum of vectors
AB and
BC is the vector
AC. It is shown in
Figure 4.
Figure 4. The sum of the vectors AB + BC = AC (the triangle rule)
 Figure 5. The sum of the vectors AB + AD = AC (the parallelogram rule)
 Figure 6. The sum of the vectors AB + EF = AB + BC = AB + AD = AC

This is the
triangle rule of adding vectors.
If the vectors AB and BC are two sides of the triangle ABC, then the sum
of the vectors AB and BC (in this order and orientation) is the third side AC of the triangle.
Now, there is the
parallelogram rule of adding vectors.
If two vectors AB and AD have the common initial point A, then the sum
of the vectors AB and AD is the vector AC which is the diagonal of the parallelogram ADCB
built on the sides AB and AD. It is shown in
Figure 5.
At last, if there are two vectors
AB and
EF with no common initial points or endpoints, as it is shown in the
Figure 6, you can move/translate one of
the vectors, let say the vector
EF, parallel to itself in such a way to superpose its initial point with the initial point or with the endpoint of the vector
AB, and then to apply the
parallelogram rule or the
triangle rule, whichever is applicable appropriately.
Due to properties of parallelograms the final result of vectors adding does not depend on which of the two rules you apply.
I mean these two properties of parallelograms:
1) if the quadrilateral has two opposite sides congruent and parallel, then the quadrilateral is a parallelogram, and
2) in a parallelogram, each diagonal divides it in two congruent triangles.
For the mentioned properties of parallelograms see the lessons
Properties of the sides of a parallelogram and
In a parallelogram, each diagonal divides it in two congruent triangles
in this site.
The operation of adding vectors has remarkable properties.
1)
a +
b =
b +
a for any two vectors
a and
b in a plane (commutativity property of addition).
Indeed, translate the vectors
a and
b in a plane parallel to themselves to superpose their initial points and then apply the
parallelogram rule.
2) (
a +
b) +
c =
a + (
b +
c) for any three vectors
a,
b and
c in a plane (associativity property of addition).
For the proof, translate the vectors
a,
b and
c in a plane parallel to themselves to superpose the initial point of
b with the endpoint of
a and
the initial point of
c with the endpoint of
b and then apply the
triangle rule. You will get that the sum of the vectors
a +
b +
c is the vector with
the initial point of
a and the endpoint of
c after the translations, and does not depend on placing parenthesis in the sum of the three vectors.
3)
a +
0 =
a (for any vector
a adding the zero vector to
a does not change the vector).
It follows the definition immediately.
4)
a +
(a) =
0 for any vector
a in a plane (existence an opposite vector).
It follows the definition immediately.
Definition of subtracting vectors in a plane
If the vectors
AB and
AC have the common initial point
A then the difference of the vectors
AB and
AC is the vector
CB which initial point is the endpoint of the vector
AC and endpoint is the endpoint of the vector
AB. It is shown in
Figure 7.
Figure 7. The difference of the vectors AB  AC = CB
 Figure 8. b + (ab) = a

The operation of subtracting vectors has remarkable properties.
1)
b + (
ab) =
a for any two vectors
a and
b.
For the proof, use the
Figure 8.
Definition the product of a vector by a real number
The product of a vector
a by a real number
r is the vector r
a whose length is equal to r
a = r*
a and whose direction coincides with that of the vector
a if r >= 0 or is opposite to the direction of the vector
a if r < 0.
If r=0 or
a=0 then the product r
a is the zero vector, i.e. its length is zero and the direction is undetermined.
The operation of product of a vector by a real number has the following properties.
1) r(
a +
b) = r
a + r
b for any vectors
a and
b and for any real number r (distributivity of product with respect to addition of vectors).
You can prove this formula using the similarity properties of triangles.
2) (r+s)
a = r
a + s
a for any vector
a and for any real numbers r and s (distributivity of product with respect to addition of constants).
3) r(s
a) = (rs)
a for any vector
a and for any real numbers r and s.
4) 1
a =
a for any vector
a (multiplication by unity).
For examples of solved problems on summing vectors see the lessons
 Sum of the vectors that are coherently oriented sides of a convex closed polygon,
 Sum of the vectors that are coherently oriented sides of an unclosed polygon, and
 Sum of the vectors that connect the center of a parallelogram with its vertices
under the current topic in this site.
My introductory lessons on vectors in this site are
 Vectors in a plane (this lesson)
 Sum of vectors that are coherently oriented sides of a convex closed polygon
 Sum of vectors that are coherently oriented sides of an unclosed polygon
 Sum of vectors that connect the center of a parallelogram with its vertices
 Vectors in a coordinate plane
 Addition, Subtraction and Multiplication by a number of vectors in a coordinate plane
 Summing vectors that are coherently oriented sides of a convex closed polygon
 Summing vectors that are coherently oriented sides of an unclosed polygon
 The Centroid of a triangle is the Intersection point of its medians
 The Centroid of a parallelogram is the Intersection point of its diagonals
 Sum of vectors connecting the center of mass of a triangle with its vertices
 Sum of vectors connecting the center of mass of a quadrilateral with its vertices
 Sum of vectors connecting the center of mass of a nsided polygon with its vertices
 Sum of vectors connecting the center of a regular nsided polygon with its vertices
 Solved problems on vectors in a plane
 Solved problems on vectors in a coordinate plane
 HOW TO find the length of the vector in a coordinate plane
Use this file/link ALGEBRAII  YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRAII.
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