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

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Summing vectors that are coherently oriented sides of an unclosed polygon


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  Sum of the vectors that are coherently oriented sides of an unclosed 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 two vectors  PQ,  QR  that are 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  PR  has the projections  x3-x1, y3-y1.

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


  Figure 1.  The triangle PQR
and the vectors PQ, QR and RP
(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   PQ +  QR.
The  x- component of the sum will be   (x2-x1) + (x3-x2) = (x3-x1).
The  y- component of the sum will be   (y2-y1) + (y3-y2) = (y3-y1.
Note that  x- and  y- components of the sum of vectors   PQ +  QR   are equal to the  x- and  y- components of the vector  PR  respectively.
Hence the sum   PQ +  QR   is equal to the vector  PR.

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 terminal point of the second additive vector.


Problem 2

Let  PQRS  be a convex quadrilateral in a coordinate plane  (Figure 2).
Find the sum of the three vectors  PQ,  QR  and  RS  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.

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  PS  has the projections  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)


Figure 2. The quadrilateral PQRS
and the vectors PQ, QR, RS and SP
(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   PQ +  QR +  RS.
The  x- component of the sum will be   (x2-x1) + (x3-x2) + (x4-x3).
Regroup the terms and then cancel the like terms with the opposite signs.  You will get that the sum is equal to  x4-x1.
The  y- component of the sum will be   (y2-y1) + (y3-y2) + (y4-y3).
Regroup the terms and then cancel the like terms with the opposite signs.  You will get that the sum is equal to  y4-y1.
Thus  x- and  y- components of the sum of vectors   PQ +  QR +  RS   are equal to the  x- and  y- components of the vector  PS  respectively.
Hence the sum   PQ +  QR +  RS   is equal to the vector  PS.

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 terminal point of the last additive vector.

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


Problem 3

Let  A1, A2, A3, . . . , An   be  n  points in a coordinate 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  (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  A1An  has the projections  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)


Figure 3. The unclosed convex polygon A1A2A3...A6
and the vectors A1A2, A2A3, A3A4, A4A5, A5A6, A1A6
(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.
The  x- component of the sum will be   (x2-x1) + (x3-x2) + (x4-x3) + . . . + (x(n)-x(n-1)).
Regroup the terms and then cancel the like terms with the opposite signs.  You will get the value of  x(n)-x(1)  for this sum.
The  y- component of the sum will be   (y2-y1) + (y3-y2) + (y4-y3) + . . . + (y(n)-y(n-1)).
Regroup the terms and then cancel the like terms with the opposite signs.  You will get the value of  y(n)-y(1)  for this sum.
Thus  x- and  y- components of the sum of vectors   A1A2 +  A2A3 +  A3A4 + . . . +  A(n-1)An   are equal to the  x- and  y- components of the sum of vector  A1An.
Hence the sum   A1A2 +  A2A3 +  A3A4 + . . . +  A(n-1)An   is equal to the vector  A1An.

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.


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
    - Summing vectors that are coherently oriented sides of an unclosed polygon                               (this lesson)
    - 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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