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

Source code of 'Summing vectors that are coherently oriented sides of an unclosed polygon'

This Lesson (Summing vectors that are coherently oriented sides of an unclosed polygon) was created by by ikleyn(52781)

: View Source, Show
About ikleyn:

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

In this lesson you will learn how to sum vectors in a coordinate plane that are coherently oriented sides of an unclosed polygon.

These problems were just solved in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-are-coherently-oriented-sides-of-an-unclosed-polygon.lesson>Sum of the vectors that are coherently oriented sides of an unclosed polygon</A>&nbsp; using the general definitions and properties of vectors in a plane,

and you can find the corresponding solutions there. &nbsp;In the current lesson the other approach is applied. &nbsp;It is based on the formula for summing vectors in the component form in a coordinate plane.

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

<Table>
 <TR>
 <TD>
Solution

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

Then the vector &nbsp;PQ&nbsp; has the projections &nbsp;x2-x1, y2-y1&nbsp; in axes &nbsp;x- and &nbsp;y- respectively,

the vector &nbsp;QR&nbsp; has the projections &nbsp;x3-x2, y3-y2, &nbsp;and 
the vector &nbsp;PR&nbsp; has the projections &nbsp;x3-x1, y3-y1.

So, you can write the vectors in the component form
PQ = (x2-x1,y2-y1),
QR = (x3-x2,y3-y2), &nbsp;and 
PR = (x3-x1,y3-y1)

</TD>
 <TD>
{{{drawing( 222, 200, -2.0, 9.1, -2.0, 8.0, 
 grid(1),
 line( 2.0, 2.0, 5.0, 6.0),
 locate ( 1.6, 2.0, P), 
 locate ( 4.6, 6.7, Q),
 line( 5.0, 6.0, 4.5, 5.7),
 line( 5.0, 6.0, 4.9, 5.5),

red(line( 2.0, 2.0, 8.0, 3.0)),
            locate (8.1,  3.5, R),

line(  8.0, 3.0,   7.5,  3.10),
            line(  8.0, 3.0,   7.45, 2.75),

line( 5.0,  6.0,  8.0,  3.0),
            line( 8.0,  3.0,  7.85, 3.5),
            line( 8.0,  3.0,  7.35, 3.3)
)}}}

&nbsp;&nbsp;Figure 1. &nbsp;The triangle PQR 
and the vectors PQ, QR and RP 
 </TD>
 </TR>
</Table>(see the lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>&nbsp; and <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> 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 &nbsp;&nbsp;PQ + &nbsp;QR.

The &nbsp;x- component of the sum will be &nbsp;&nbsp;(x2-x1) + (x3-x2) = (x3-x1).
The &nbsp;y- component of the sum will be &nbsp;&nbsp;(y2-y1) + (y3-y2) = (y3-y1.

Note that &nbsp;x- and &nbsp;y- components of the sum of vectors &nbsp;&nbsp;PQ + &nbsp;QR &nbsp;&nbsp;are equal to the &nbsp;x- and &nbsp;y- components of the vector &nbsp;PR&nbsp; respectively.

Hence the sum &nbsp;&nbsp;PQ + &nbsp;QR &nbsp;&nbsp;is equal to the vector &nbsp;PR.

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 terminal point of the second additive vector.

<H3>Problem 2</H3>Let &nbsp;PQRS&nbsp; be a convex quadrilateral in a coordinate plane &nbsp;(Figure 2).
Find the sum of the three vectors &nbsp;PQ, &nbsp;QR&nbsp; and &nbsp;RS&nbsp; that are the coherently oriented consecutive sides of the quadrilateral.

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

<Table>
 <TR>
 <TD>
Solution

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

Then the vector &nbsp;PQ&nbsp; has the projections &nbsp;x2-x1, y2-y1&nbsp; in axes &nbsp;x- and &nbsp;y- respectively,

the vector &nbsp;QR&nbsp; has the projections &nbsp;x3-x2, y3-y2, 
the vector &nbsp;RS&nbsp; has the projections &nbsp;x4-x3, y4-y3, &nbsp;and 
the vector &nbsp;PS&nbsp; has the projections &nbsp;x4-x1, y4-y1.

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)
PS = (x4-x1,y4-y1)

</TD>
 <TD>
{{{drawing( 282, 240, -2.0, 12.1, -2.0, 10.0,
 grid(1), 
 line( 2.0, 2.0, 5.0, 6.0),
 locate ( 1.8, 2.0, P), 
 locate ( 4.64, 6.7, Q),
 line( 5.0, 6.0, 4.5, 5.7),
 line( 5.0, 6.0, 4.9, 5.5),

line( 5.0, 6.0, 11.0, 7.0),
            locate (11.1,   7.5, R),

line( 11.0, 7.0,  10.35, 7.1), 
            line( 11.0, 7.0,  10.35, 6.65),

red(line( 2.0, 2.0, 11.0, 2.0)),
            locate (10.8,   2.0, S),
            line( 11.0, 2.0,  11.20, 2.5),
            line( 11.0, 2.0,  10.80, 2.5),

line(11.0, 2.0, 11.0, 7.0),

line( 11.0, 2.0, 10.40, 2.12),
 line( 11.0, 2.0, 10.40, 1.85)
)}}}
Figure 2. The quadrilateral PQRS 
and the vectors PQ, QR, RS and SP
 </TD>
 </TR>
</Table>(see the lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>&nbsp; and <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> 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 &nbsp;&nbsp;PQ + &nbsp;QR + &nbsp;RS.

The &nbsp;x- component of the sum will be &nbsp;&nbsp;(x2-x1) + (x3-x2) + (x4-x3). 
Regroup the terms and then cancel the like terms with the opposite signs. &nbsp;You will get that the sum is equal to &nbsp;x4-x1.

The &nbsp;y- component of the sum will be &nbsp;&nbsp;(y2-y1) + (y3-y2) + (y4-y3). 
Regroup the terms and then cancel the like terms with the opposite signs. &nbsp;You will get that the sum is equal to &nbsp;y4-y1.

Thus &nbsp;x- and &nbsp;y- components of the sum of vectors &nbsp;&nbsp;PQ + &nbsp;QR + &nbsp;RS&nbsp;&nbsp; are equal to the &nbsp;x- and &nbsp;y- components of the vector &nbsp;PS&nbsp; respectively.

Hence the sum &nbsp;&nbsp;PQ + &nbsp;QR + &nbsp;RS&nbsp;&nbsp; is equal to the vector &nbsp;PS.

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 terminal point of the last additive vector.

Note. &nbsp;It is clear from the solution that the same result is valid for non-convex quadrilaterals too.

<H3>Problem 3</H3>Let &nbsp;A1, A2, A3, . . . , An &nbsp; be &nbsp;n&nbsp; points in a coordinate 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>
 <TR>
 <TD>
Solution

Let &nbsp;(x1,y1)&nbsp; be the coordinates of the point &nbsp;A1 in a coordinate plane, &nbsp;A1 = A1(x1,y1),&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
(x2,y2)&nbsp; be the coordinates of the point &nbsp;A2, &nbsp;A2 = A2(x2,y2), 
(x3,y3)&nbsp; be the coordinates of the point &nbsp;A3, &nbsp;A3 = A3(x3,y3), &nbsp; 
. . . . . . . .

(x(n-1),y(n-1))&nbsp; be the coordinates of the point &nbsp;A(n-1), &nbsp;A(n-1) = A(n-1)(x(n-1),y(n-1)), &nbsp;and 
(x(n),y(n))&nbsp; be the coordinates of the point &nbsp;An, &nbsp;An = An(x(n),y(n)).

Then the vector &nbsp;A1A2&nbsp; has the projections &nbsp;x2-x1, y2-y1&nbsp; in axes &nbsp;x- and &nbsp;y- respectively,

the vector &nbsp;A2A3&nbsp; has the projections &nbsp;x3-x2, y3-y2, 
the vector &nbsp;A3A4&nbsp; has the projections &nbsp;x4-x3, y4-y3, &nbsp;and so on. 
The two last vectors are:
the vector &nbsp;A(n-1)An&nbsp; has the projections &nbsp;x(n)-x(n-1), y(n)-y(n-1), &nbsp;and 
the vector &nbsp;A1An&nbsp; has the projections &nbsp;x(n)-x(1), y(n)-y(1).

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)),
A1An = (x(n)-x(1)),y(n)-y1)

</TD>
 <TD>
{{{drawing( 322, 280, -2.0, 14.1, -2.0, 12.0,
 grid(1), 
 line( 3.0, 2.0, 2.0, 4.0),
 locate ( 2.7, 1.8, A1), 
 locate ( 1.1, 4.4, A2),
 line( 2.0, 4.0, 2.1, 3.5),
 line( 2.0, 4.0, 2.4, 3.6),

line( 2.0, 4.0,  4.0, 8.0),
            locate ( 3.2, 8.7, A3), 
            line(  4.0,  8.0,  3.6, 7.55),
            line(  4.0,  8.0,  3.9, 7.40),

line( 4.0,  8.0,  8.0, 9.0),
            locate (7.6, 9.9, A4), 
            line(  8.0,  9.0,  7.6, 9.10),
            line(  8.0,  9.0,  7.6, 8.70),

line( 8.0,  8.9,  11.0, 7.0),
            locate (11.1, 7.7, A5),
            line(  11.0,  7.0,  10.75, 7.40),
            line(  11.0,  7.0,  10.40, 7.10),

line( 11.0,   7.0,  12.0, 2.0),
            locate (11.7, 1.8, A6),
            line(  12.0, 2.0,  11.65, 2.40),
            line(  12.0, 2.0,  12.15, 2.50),

red(line( 3.0, 1.9, 12.0, 2.0)),
 line( 12.0, 2.0, 11.60, 2.10),
 line( 12.0, 2.0, 11.60, 1.77)
)}}}
Figure 3. The unclosed convex polygon A1A2A3...A6
and the vectors A1A2, A2A3, A3A4, A4A5, A5A6, A1A6
 </TD>
 </TR>
</Table>(see the lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>&nbsp; and <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> 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 &nbsp;&nbsp;A1A2 + &nbsp;A2A3 + &nbsp;A3A4 + . . . + &nbsp;A(n-1)An.

The &nbsp;x- component of the sum will be &nbsp;&nbsp;(x2-x1) + (x3-x2) + (x4-x3) + . . . + (x(n)-x(n-1)).

Regroup the terms and then cancel the like terms with the opposite signs. &nbsp;You will get the value of &nbsp;x(n)-x(1)&nbsp; for this sum.

The &nbsp;y- component of the sum will be &nbsp;&nbsp;(y2-y1) + (y3-y2) + (y4-y3) + . . . + (y(n)-y(n-1)).

Regroup the terms and then cancel the like terms with the opposite signs. &nbsp;You will get the value of &nbsp;y(n)-y(1)&nbsp; for this sum.

Thus &nbsp;x- and &nbsp;y- components of the sum of vectors &nbsp;&nbsp;A1A2 + &nbsp;A2A3 + &nbsp;A3A4 + . . . + &nbsp;A(n-1)An&nbsp;&nbsp; are equal to the &nbsp;x- and &nbsp;y- components of the sum of vector &nbsp;A1An. 
Hence the sum &nbsp;&nbsp;A1A2 + &nbsp;A2A3 + &nbsp;A3A4 + . . . + &nbsp;A(n-1)An&nbsp;&nbsp; is equal to the vector &nbsp;A1An.

Answer. &nbsp;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.

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

&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/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;&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>

&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.