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

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<H2>Sum of vectors that are coherently oriented sides of an unclosed polygon</H2>

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

<H3>Problem 1</H3>Let &nbsp;ABC&nbsp; be a triangle in a plane &nbsp;(Figure 1).
Find the sum of the two vectors &nbsp;AB, &nbsp;BC&nbsp; that are coherently oriented consecutive sides of the triangle.

<Table>
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Solution

According to the definition of adding vectors &nbsp;(the triangle rule, &nbsp;see the lesson

<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>&nbsp; under the current topic in this site), &nbsp;the sum of the two vectors&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

AB&nbsp; and &nbsp;BC&nbsp; is the vector &nbsp;AC, &nbsp;which represents the third side of the triangle &nbsp;ABC.

The initial point of this vector is the initial point &nbsp;A&nbsp; of the first additive, &nbsp;vector &nbsp;AB, 
while its ending point is the ending point &nbsp;C&nbsp; of the second additive, &nbsp;vector &nbsp;BC:

AB + BC = AC.

Answer. &nbsp;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.
 </TD>
 <TD>
{{{drawing( 222, 140, -6.0, 5.1, -4.0, 3.0,
 line(-5.0, -3.0, -2.0, 1.0),
 locate (-5.2, -3.0, A), 
 locate (-2.4, 1.7, B),
 line( -2.0, 1.0, -2.5, 0.7),
 line( -2.0, 1.0, -2.1, 0.5),

line(-2.0, 1.0, 1.0, -2.0),
            line( 1.0, -2.0,  0.85, -1.5),
            line( 1.0, -2.0,  0.35, -1.7),

line(-5.0, -3.0, 1.0, -2.0),
            locate (1.1,  -1.5, C),
            line( 1.0, -2.0, 0.4, -1.95),
            line( 1.0, -2.0, 0.4, -2.27)
)}}}

&nbsp;&nbsp;Figure 1. &nbsp;The triangle ABC 
and the vectors AB, BC and AC 
 </TD>
 </TR>
</Table>

<H3>Problem 2</H3>Let &nbsp;ABCD&nbsp; be a convex quadrilateral in a plane &nbsp;(Figure 2).
Find the sum of the three vectors &nbsp;AB, &nbsp;BC&nbsp; and &nbsp;CD&nbsp; that are the coherently oriented consecutive sides of the quadrilateral.

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 <TR>
 <TD>
Solution

Let us draw the diagonal &nbsp;AC&nbsp; in the quadrilateral &nbsp;ABCD.
According to the definition of adding vectors &nbsp;(the triangle rule, &nbsp;see the lesson

<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A> under the current topic in this site), &nbsp;the sum of the two vectors&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

AB&nbsp; and &nbsp;BC &nbsp;is the vector &nbsp;AC. &nbsp;Hence, in the sum of the three vectors &nbsp;AB + BC + CD

we can replace the sum of the first two vectors by the vector &nbsp;AC:

AB + BC + CD = AC + CD.

The right side of this equality is the sum of the two vectors that are 
 </TD>
 <TD>
{{{drawing( 222, 140, -6.0, 5.1, -4.0, 3.0,
 line(-5.0, -3.0, -2.0, 1.0),
 locate (-5.2, -3.0, A), 
 locate (-2.4, 1.7, B),
 line( -2.0, 1.0, -2.5, 0.7),
 line( -2.0, 1.0, -2.1, 0.5),

line(-2.0, 1.0, 4.0, 2.0),
            locate (4.1,   2.5, C),

red(line(-5.0, -3.0, 4.0, 1.95)),
            line( 4.0, 1.95,  3.5, 1.52), 
            line( 4.0, 1.95,  3.3, 1.75),

line( 4.0, 2.0,  3.35, 2.05),

line(-5.0, -3.0, 4.0, -3.0),
            locate (3.8,  -3.0, D),
            line( 4.0, -3.0,  4.20, -2.5),
            line( 4.0, -3.0,  3.80, -2.5),

line(4.0, -3.0, 4.0, 2.0),

line( 4.0, -3.0, 3.40, -2.88),
 line( 4.0, -3.0, 3.40, -3.15)
)}}}
Figure 2. The quadrilateral ABCD 
and the vectors AB, BC, CD and AD
 </TD>
 </TR>
</Table>coherently oriented consecutive sides of the triangle &nbsp;ACD. &nbsp;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. &nbsp;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.

<H3>Problem 3</H3>Let &nbsp;A1, A2, A3, . . . , An &nbsp; be &nbsp;n&nbsp; points in a plane &nbsp;(n >= 3)&nbsp; and &nbsp;A1A2A3...An&nbsp; be a line in the plane consisting of &nbsp;n-1&nbsp; straight line segments that connect the points &nbsp;A1 and A2, &nbsp;A2 and A3, . . . , A(n-1) and An, &nbsp;respectively. &nbsp;The example of such a line is shown in &nbsp;Figure 3&nbsp; for n=6. &nbsp;Note that this line is an unclosed polygon.

Find the sum of the &nbsp;n-1&nbsp; vectors &nbsp;A1A2, A2A3, A3A4, . . . , A(n-1)An.

<Table>
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 <TD>
Solution

Let us draw the straight line segment &nbsp;A1A3&nbsp; connecting the points A1 and A3 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
(Figure 4). According to the definition of adding vectors &nbsp;(the triangle rule,

see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>&nbsp; under the current topic in this site),

the sum of the two vectors &nbsp;A1A2&nbsp; and &nbsp;A2A3&nbsp; is the vector &nbsp;A1A3. &nbsp;Hence,

in the sum of &nbsp;n-1 &nbsp;vectors &nbsp;A1A2 + A2A3 + A3A4 + . . . + A(n-1)An

we can replace the sum of the first two vectors by the vector &nbsp;A1A3:

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

The right side of this equality is the sum of &nbsp;n-2&nbsp; vectors that are 
coherently oriented straight segments connecting &nbsp;n-2&nbsp; points &nbsp;A1, A3, . . . , An
 </TD>
 <TD>
{{{drawing( 242, 180, -6.0, 6.1, -4.0, 5.0,
 line(-4.0, -3.0, -5.0, -1.0),
 locate (-4.3, -3.2, A1), 
 locate (-5.9, -0.6, A2),
 line( -5.0, -1.0, -4.9, -1.5),
 line( -5.0, -1.0, -4.6, -1.4),

line(-5.0, -1.0, -3.0, 3.0),
            locate (-3.8, 3.7, A3), 
            line( -3.0,  3.0, -3.4, 2.55),
            line( -3.0,  3.0, -3.1, 2.40),

line(-3.0,  3.0,  1.0, 4.0),
            locate (0.6, 4.9, A4), 
            line(  1.0,  4.0,  0.6, 4.10),
            line(  1.0,  4.0,  0.6, 3.70),

line( 1.0,  3.9,  4.0, 2.0),
            locate (4.1, 2.7, A5),
            line(  4.0,  2.0,  3.75, 2.40),
            line(  4.0,  2.0,  3.40, 2.10),

line( 4.0, 2.0, 5.0, -3.0),
 locate (4.7, -3.2, A6),
 line( 5.0, -3.0, 4.65, -2.60),
 line( 5.0, -3.0, 5.15, -2.50)
)}}}
Figure 3. The points A1, A2, A3, ... , A6,&nbsp;&nbsp;&nbsp;&nbsp; 
the line A1A2A3...A6 and the vectors 
A1A2, A2A3, ... , A5A6
 </TD>
 <TD>
{{{drawing( 242, 180, -6.0, 6.1, -4.0, 5.0,
 line(-4.0, -3.0, -5.0, -1.0),
 locate (-4.3, -3.2, A1), 
 locate (-5.9, -0.6, A2),
 line( -5.0, -1.0, -4.9, -1.5),
 line( -5.0, -1.0, -4.6, -1.4),

line(-5.0, -1.0, -3.0, 3.0),
            locate (-3.8, 3.7, A3), 
            line( -3.0,  3.0, -3.4, 2.55),
            line( -3.0,  3.0, -3.1, 2.40),

line(-3.0,  3.0,  1.0, 4.0),
            locate (0.6, 4.9, A4), 
            line(  1.0,  4.0,  0.6, 4.10),
            line(  1.0,  4.0,  0.6, 3.70),

line( 1.0,  3.9,  4.0, 2.0),
            locate (4.1, 2.7, A5),
            line(  4.0,  2.0,  3.75, 2.40),
            line(  4.0,  2.0,  3.40, 2.10),

line( 4.0,  2.0,  5.0, -3.0),
            locate (4.7, -3.2, A6),
            line(  5.0, -3.0,  4.65, -2.60),
            line(  5.0, -3.0,  5.15, -2.50),

red(line(-4.0, -3.0,  -3.0, 3.0)),
        red(line(-4.0, -3.0,   1.0, 4.0)),
        red(line(-4.0, -3.0,   4.0, 2.0)),

red(line( -4.0, -3.1,  5.0, -3.0)),
            line(  5.0, -3.0,  4.60, -2.90),
            line(  5.0, -3.0,  4.60, -3.23)

)}}}
Figure 4. The unclosed polygon A1A2A3...A6, 
the intermediate diagonals A1A3, A1A4, ... ,
and the final sum, &nbsp;vector &nbsp;A1A6
 </TD>
 </TR>
</Table>(the point &nbsp;A2&nbsp; is omitted now). &nbsp;Hence, &nbsp;we managed to decrease the number of points from &nbsp;n&nbsp; to &nbsp;n-1&nbsp; and the number of straight line segments from &nbsp;n-1&nbsp; to &nbsp;n-2, &nbsp;respectively, with no change the required&nbsp; (the original)&nbsp; sum of the vectors.

Now we can repeat this step one more time. &nbsp;Let us draw the straight line segment &nbsp;A1A4&nbsp; in the unclosed polygon &nbsp;A1A3A4...An&nbsp; connecting the vertices &nbsp;A1&nbsp; and &nbsp;A4.&nbsp; According to the definition &nbsp;(the triangle rule, &nbsp;see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>&nbsp; under the current topic in this site), &nbsp;the sum of the two vectors&nbsp; A1A3&nbsp; and &nbsp;A3A4&nbsp; is the vector &nbsp;A1A4. &nbsp;Hence, &nbsp;in the sum of &nbsp;n-2&nbsp; vectors &nbsp;A1A3 + A3A4 + A4A5 + . . . + A(n-1)An&nbsp; we can replace the sum of the first two vectors by the vector &nbsp;A1A4&nbsp; 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 &nbsp;n-3&nbsp; vectors that are coherently oriented consecutive sides of the unclosed polygon &nbsp;A1A4A5...An&nbsp; with &nbsp;n-2&nbsp; vertices and &nbsp;n-3&nbsp; sides. &nbsp;Hence, &nbsp;this time we decreased the number of vertices of the unclosed polygon from &nbsp;n-1&nbsp; to &nbsp;n-2&nbsp; and the number of sides from &nbsp;n-2&nbsp; to &nbsp;n-3&nbsp; with no change the required (original) sum of vectors.

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

Answer. For any unclosed &nbsp;(n-1)-sided polygon the sum of &nbsp;(n-1)&nbsp; 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. &nbsp;This vector closes the original unclosed polygon.

For other examples of solved problems on summing vectors see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-are-coherently-oriented-sides-of-a-convex-closed-polygon.lesson>Sum of the vectors that are coherently oriented sides of a convex closed polygon</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-connect-the-center-of-a-parallelogram-with-its-vertices.lesson>Sum of the vectors that connect the center of a parallelogram with its vertices</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-from-the-center-of-mass-of-the-n-sided-polygon-to-its-vertices.lesson>Sum of vectors connecting the center of mass of a n-sided polygon with its vertices</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-vectors-connecting-the-center-of-a-regular-polygon-with-its-vertices.lesson>Sum of vectors connecting the center of a regular n-sided polygon with its vertices</A>

under the current topic in this site.

My introductory lessons on vectors in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-are-coherently-oriented-sides-of-a-convex-closed-polygon.lesson>Sum of vectors that are coherently oriented sides of a convex closed polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- Sum of vectors that are coherently oriented sides of an unclosed polygon&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(this lesson)

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-connect-the-center-of-a-parallelogram-with-its-vertices.lesson>Sum of vectors that connect the center of a parallelogram with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Addition-Subtraction-and-Multiplication-by-a-number-of-vectors-in-a-coordinate-plane.lesson>Addition, Subtraction and Multiplication by a number of vectors in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Summing-vectors-that-are-coherently-oriented-sides-of-a-convex-closed-polygon.lesson>Summing vectors that are coherently oriented sides of a convex closed polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Summing-vectors-that-are-coherently-oriented-sides-of-an-unclosed-polygon.lesson>Summing vectors that are coherently oriented sides of an unclosed polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-Centroid-of-a-triangle-is-the-Intersection-point-of-its-medians.lesson>The Centroid of a triangle is the Intersection point of its medians</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-Centroid-of-a-parallelogram-is-the-Intersection-point-of-its-diagonals.lesson>The Centroid of a parallelogram is the Intersection point of its diagonals</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Sum-of-the-vectors-connecting-the-center-of-mass-of-a-triangle-with-its-vertices.lesson>Sum of vectors connecting the center of mass of a triangle with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Sum-of-the-vectors-connecting-the-center-of-mass-of-a-quadrilateral-with-its-vertices.lesson>Sum of vectors connecting the center of mass of a quadrilateral with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Solved-problems-on-vectors-in-a-plane.lesson>Solved problems on vectors in a plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Solved-problems-on-vectors-in-a-coordinate-plane.lesson>Solved problems on vectors in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-find-the-length-of-the-vector-in-a-coordinate-plane.lesson>HOW TO find the length of the vector in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Vectors/Flying-airplane-blowing-wind-airspeed-groundspeed-etc.lesson>Flying airplane, blowing wind, airspeed, groundspeed etc.</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Vectors/REVIEW-of-Introductory-lessons-on-vectors-in-a-plane.lesson>OVERVIEW of Introductory lessons on vectors in a plane</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.