Lesson Sum of vectors that are coherently oriented sides of an unclosed polygon

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Sum of vectors that are coherently oriented sides of an unclosed polygon

In this lesson you will learn about summing the vectors that are coherently oriented sides of an unclosed polygon.

Problem 1

Let ABC be a triangle in a plane (Figure 1).
Find the sum of the two vectors AB, BC that are coherently oriented consecutive sides of the triangle.

Solution
According to the definition of adding vectors (the triangle rule, see the lesson
Vectors in a plane under the current topic in this site), the sum of the two vectors
AB and BC is the vector AC, which represents the third side of the triangle ABC.
The initial point of this vector is the initial point A of the first additive, vector AB,
while its ending point is the ending point C of the second additive, vector BC:

AB + BC = AC.

Answer. The sum of two vectors that are coherently oriented consecutive
sides of a triangle is the third side of the triangle oriented from the initial point
of the first additive vector to the endpoint of the second additive vector.

Figure 1. The triangle ABC
and the vectors AB, BC and AC

Problem 2

Let ABCD be a convex quadrilateral in a plane (Figure 2).
Find the sum of the three vectors AB, BC and CD that are the coherently oriented consecutive sides of the quadrilateral.

Solution
Let us draw the diagonal AC in the quadrilateral ABCD.
According to the definition of adding vectors (the triangle rule, see the lesson
Vectors in a plane under the current topic in this site), the sum of the two vectors
AB and BC is the vector AC. Hence, in the sum of the three vectors AB + BC + CD
we can replace the sum of the first two vectors by the vector AC:

AB + BC + CD = AC + CD.

The right side of this equality is the sum of the two vectors that are

Figure 2. The quadrilateral ABCD
and the vectors AB, BC, CD and AD

coherently oriented consecutive sides of the triangle ACD. Such a sum is equal to the third side of the triangle oriented from the initial point of the first additive vector to the endpoint of the second additive vector. It is nothing else as the fourth side of the quadrilateral oriented from the initial point of the first additive vector to the endpoint of the third additive vector in the original sum of three vectors:

AB + BC + CD = AD.

Answer. The sum of three vectors that are coherently oriented consecutive sides of a quadrilateral is the fourth side of the quadrilateral oriented from the initial point of the first additive vector to the endpoint of the last additive vector.

Problem 3

Let A1, A2, A3, . . . , An be n points in a plane (n >= 3) and A1A2A3...An be a line in the plane consisting of n-1 straight line segments that connect the points A1 and A2, A2 and A3, . . . , A(n-1) and An, respectively. The example of such a line is shown in Figure 3 for n=6. Note that this line is an unclosed polygon.
Find the sum of the n-1 vectors A1A2, A2A3, A3A4, . . . , A(n-1)An.

Solution
Let us draw the straight line segment A1A3 connecting the points A1 and A3
(Figure 4). According to the definition of adding vectors (the triangle rule,
see the lesson Vectors in a plane under the current topic in this site),
the sum of the two vectors A1A2 and A2A3 is the vector A1A3. Hence,
in the sum of n-1 vectors A1A2 + A2A3 + A3A4 + . . . + A(n-1)An
we can replace the sum of the first two vectors by the vector A1A3:

A1A2 + A2A3 + A3A4 + . . . + A(n-1)An = A1A3 + A3A4 + . . . + A(n-1)An.

The right side of this equality is the sum of n-2 vectors that are
coherently oriented straight segments connecting n-2 points A1, A3, . . . , An

Figure 3. The points A1, A2, A3, ... , A6,
the line A1A2A3...A6 and the vectors
A1A2, A2A3, ... , A5A6

Figure 4. The unclosed polygon A1A2A3...A6,
the intermediate diagonals A1A3, A1A4, ... ,
and the final sum, vector A1A6

(the point A2 is omitted now). Hence, we managed to decrease the number of points from n to n-1 and the number of straight line segments from n-1 to n-2, respectively, with no change the required (the original) sum of the vectors.

Now we can repeat this step one more time. Let us draw the straight line segment A1A4 in the unclosed polygon A1A3A4...An connecting the vertices A1 and A4. According to the definition (the triangle rule, see the lesson Vectors in a plane under the current topic in this site), the sum of the two vectors A1A3 and A3A4 is the vector A1A4. Hence, in the sum of n-2 vectors A1A3 + A3A4 + A4A5 + . . . + A(n-1)An we can replace the sum of the first two vectors by the vector A1A4 which is our current diagonal:

A1A3 + A3A4 + A4A5 + . . . + A(n-1)An = A1A4 + A3A4 + . . . + A(n-1)An.

This time the right side of the last equality is the sum of n-3 vectors that are coherently oriented consecutive sides of the unclosed polygon A1A4A5...An with n-2 vertices and n-3 sides. Hence, this time we decreased the number of vertices of the unclosed polygon from n-1 to n-2 and the number of sides from n-2 to n-3 with no change the required (original) sum of vectors.

Repeating this method n-2 times (by the number of diagonals) we will get finally the case of the three points A1, A(n-1), An, and the two vectors A1A(n-1) and A(n-1)An. Our original sum of the vectors will be equal to the sum of the two vectors A1A(n-1) (the last diagonal connecting vertices A1 and A(n-1)) and A(n-1)An. According to the definition of adding vectors (the triangle rule, see the lesson Vectors in a plane under the current topic in this site), the sum of the vectors A1A(n-1) and A(n-1)An is the vector connecting the starting point of the first additive vector and the endpoint of the second additive vector, i.e. the vector A1An . Hence, our original sum of (n-1) vectors for the (n-1)-sided unclosed polygon A1A2A3...A(n-1)An is equal to the vector A1An which closes the original polygon line.

Answer. For any unclosed (n-1)-sided polygon the sum of (n-1) vectors that are the coherently oriented sides of the polygon is the vector which connects the initial point of the first additive vector with the endpoint of the last additive vector and is oriented accordingly. This vector closes the original unclosed polygon.

For other 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 connect the center of a parallelogram with its vertices,
    - Sum of vectors connecting the center of mass of a n-sided polygon with its vertices and
    - Sum of vectors connecting the center of a regular n-sided polygon with its vertices
under the current topic in this site.

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                                 (this lesson)
    - 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 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.

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