Lesson Summing vectors that are coherently oriented sides of a convex closed polygon

This Lesson (Summing vectors that are coherently oriented sides of a convex closed polygon) was created by by ikleyn(52781)

: View Source, Show
About ikleyn:

Summing vectors that are coherently oriented sides of a convex closed polygon

In this lesson you will learn how to sum vectors in a coordinate plane that are coherently oriented sides of a convex closed polygon.
These problems were just solved in the lesson Sum of the vectors that are coherently oriented sides of a convex closed polygon using the general definitions and properties of vectors in a plane,
and you can find the corresponding solutions there. In the current lesson the other approach is applied. It is based on the formula for summing vectors in the component form in a coordinate plane.

Problem 1

Let PQR be a triangle in a coordinate plane (Figure 1).
Find the sum of the vectors PQ, QR and RP that are the coherently oriented consecutive sides of the triangle.

Solution
Let (x1,y1) be the coordinates of the point P in a coordinate plane, P = P(x1,y1),
(x2,y2) be the coordinates of the point Q, Q = Q(x2,y2), and
(x3,y3) be the coordinates of the point R, R = R(x3,y3).

Then the vector PQ has the projections x2-x1, y2-y1 in axes x- and y- respectively,
the vector QR has the projections x3-x2, y3-y2, and
the vector RP has the projections x1-x3, y1-y3.

So, you can write the vectors in the component form
PQ = (x2-x1,y2-y1),
QR = (x3-x2,y3-y2),
RP = (x1-x3,y1-y3)

Figure 1. The triangle PQR
and the vectors PQ, QR and RP

Problem 2

Let PQRS be a convex quadrilateral in a coordinate plane (Figure 2).
Find the sum of the vectors PQ, QR, RS and SP that are the coherently oriented consecutive sides of the quadrilateral.

Solution
Let (x1,y1) be the coordinates of the point P in a coordinate plane, P = P(x1,y1),
(x2,y2) be the coordinates of the point Q, Q = Q(x2,y2),
(x3,y3) be the coordinates of the point R, R = R(x3,y3), and
(x4,y4) be the coordinates of the point S, S = S(x4,y4).

Then the vector PQ has the projections x2-x1, y2-y1 in axes x- and y- respectively,
the vector QR has the projections x3-x2, y3-y2,
the vector RS has the projections x4-x3, y4-y3, and
the vector SP has the projections x1-x4, y1-y4.

So, you can write the vectors in the component form
PQ = (x2-x1,y2-y1),
QR = (x3-x2,y3-y2),
RS = (x4-x3,y4-y3)
SP = (x1-x4,y1-y4)

Figure 2. The quadrilateral PQRS
and the vectors PQ, QR, RS and SP

Problem 3

Let A1A2A3...An be a convex closed polygon in a coordinate plane with n vertices A1, A2, A3, . . . , An and n sides. It is shown in Figure 3 for n=6.
Find the sum of the vectors A1A2, A2A3, A3A4, . . . , A(n-1)An, AnA1 that are the coherently oriented consecutive sides of the polygon.

The term "coherently oriented sides of a polygon" means that for any two consecutive sides of a polygon the endpoint of the current side is the starting point of the next side.

Solution
Let (x1,y1) be the coordinates of the point A1 in a coordinate plane, A1 = A1(x1,y1),
(x2,y2) be the coordinates of the point A2, A2 = A2(x2,y2),
(x3,y3) be the coordinates of the point A3, A3 = A3(x3,y3),
. . . . . . . .
(x(n-1),y(n-1)) be the coordinates of the point A(n-1), A(n-1) = A(n-1)(x(n-1),y(n-1)), and
(x(n),y(n)) be the coordinates of the point An, An = An(x(n),y(n)).

Then the vector A1A2 has the projections x2-x1, y2-y1 in axes x- and y- respectively,
the vector A2A3 has the projections x3-x2, y3-y2,
the vector A3A4 has the projections x4-x3, y4-y3, and so on.
The two last vectors are:
the vector A(n-1)An has the projections x(n)-x(n-1), y(n)-y(n-1), and
the vector AnA1 has the projections x1-x(n), y(1)-y(n).

So, you can write the vectors in the component form
A1A2 = (x2-x1,y2-y1),
A2A3 = (x3-x2,y3-y2),
A3A4 = (x4-x3,y4-y3),
. . . . . . . .
A(n-1)An = (x(n)-x(n-1),y(n)-y(n-1)),
AnA1 = (x1-x(n),y(1)-y(n))

Figure 3. The closed convex polygon A1A2A3...A6
and the vectors A1A2, A2A3, A3A4, A4A5, A5A6, A6A1

(see the lessons Vectors in a coordinate plane and Addition, Subtraction and Multiplication by a number of vectors in a coordinate plane under the current topic in this site).

Now, apply the formula for adding vectors in a coordinate plane in the component form to calculate the sum of vectors   A1A2 + A2A3 + A3A4 + . . . + A(n-1)An + AnA1.
The x- component of the sum will be   (x2-x1) + (x3-x2) + (x4-x3) + . . . + (x(n)-x(n-1)) + (x1-x(n)).
Regroup the terms and then cancel the like terms with the opposite signs. You will get the value of zero for this sum.
The y- component of the sum will be   (y2-y1) + (y3-y2) + (y4-y3) + . . . + (y(n)-y(n-1)) + (y1-y(n)).
Regroup the terms and then cancel the like terms with the opposite signs. You will get the value of zero for this sum.
Thus both x- and y- components of the sum of vectors   A1A2 + A2A3 + A3A4 + . . . + A(n-1)An + AnA1   are equal to zero.
Hence the sum   A1A2 + A2A3 + A3A4 + . . . + A(n-1)An + AnA1   is equal to the zero vector.

Answer. The sum of the n vectors that are the coherently oriented sides of a convex n-sided polygon is equal to zero.

Note. It is clear from the solution that the same result is valid for non-convex n-sided closed polygons too.

My introductory lessons on vectors in this site are
    - Vectors in a plane
    - 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                        (this lesson)
    - 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 n-sided polygon with its vertices
    - Sum of vectors connecting the center of a regular n-sided 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
    - Flying airplane, blowing wind, airspeed, groundspeed etc.

    - OVERVIEW of Introductory lessons on vectors in a plane

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

This lesson has been accessed 2844 times.