Lesson Sum of vectors that connect the center of a parallelogram with its vertices

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<H2>Sum of vectors that connect the center of a parallelogram with its vertices</H2>

In this lesson you will learn about summing the vectors that connect the center of a parallelogram with its vertices.

<H3>Problem 1</H3>Let &nbsp;ABCD&nbsp; be a square in a plane &nbsp;(Figure 1).
Find the sum of the vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; that that connect the center &nbsp;O&nbsp; of the square with its vertices.

<Table>
 <TR>
 <TD>
Solution

A square is the particular case of a parallelogram. &nbsp;It is well known fact that the diagonals&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
of a parallelogram bisect each other, see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A>&nbsp;&nbsp;
in the section Geometry in this site. &nbsp;Hence, the diagonals of a square bisect each other. 
Let us group the sum of the four given vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; as follows:

OA + OB + OC + OD = (OA + OC) + (OB + OD).

</TD>
 <TD>
{{{drawing( 215, 180, -1.0, 5.1, -0.5, 4.5,
 line( 0.0, 0.0, 6.0, 0.0),
 locate (-0.1, -0.1, A),

line( 4.0, 0.0,  4.0, 4.0),
            locate ( 3.9, -0.1,  B),

line( 0.0, 4.0, 4.0, 4.0),
            locate ( 4.1,   4.3,  C),

line( 0.0, 0.0, 0.0, 4.0),
            locate (-0.3,   4.3,  D),

line( 0.0, 0.0, 4.0, 4.0),
            line( 4.0, 4.0, 3.8, 3.6),
            line( 4.0, 4.0, 3.6, 3.8),

line( 0.0, 0.0, 0.2, 0.4),
            line( 0.0, 0.0, 0.4, 0.2),

line( 4.0, 0.0, 0.0, 4.0),
            line( 4.0, 0.0, 3.8, 0.4),
            line( 4.0, 0.0, 3.6, 0.2),

line( 0.0, 4.0, 0.2, 3.6),
            line( 0.0, 4.0, 0.4, 3.8),

locate (1.9, 1.9, O)
)}}}
&nbsp;&nbsp;Figure 1. &nbsp;The square ABCD 
and the vectors OA, OB, OC and OD 
 </TD>
 </TR>
</Table>

Answer. &nbsp;The sum of the four vectors that connect the center of a square with its vertices is the zero vector.

<H3>Problem 2</H3>Let &nbsp;ABCD&nbsp; be a rectangular in a plane &nbsp;(Figure 2).
Find the sum of the vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; that connect the center &nbsp;O&nbsp; of the rectangular with its vertices.

<Table>
 <TR>
 <TD>
Solution

A rectangular is the particular case of a parallelogram. &nbsp;It is well known fact that the diagonals&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
of a parallelogram bisect each other, see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A>&nbsp;&nbsp;
in the section Geometry in this site. &nbsp;Hence, the diagonals of a rectangular bisect each other. 
Let us group the sum of the four given vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; as follows:

OA + OB + OC + OD = (OA + OC) + (OB + OD).

The vectors in each parenthesis are lying in one straight line, have the same length and are 
opposite. &nbsp;Hence, the sum of the vectors in each parenthesis is equal to zero. &nbsp;Therefore, 
the sum of the four given vectors is zero.
 </TD>
 <TD>
{{{drawing( 280, 180, -1.0, 7.1, -0.5, 4.5,
 line( 0.0, 0.0, 6.0, 0.0),
 locate (-0.1, -0.1, A),

line( 6.0, 0.0,  6.0, 4.0),
            locate ( 5.9, -0.1,  B),

line( 0.0, 4.0, 6.0, 4.0),
            locate ( 6.1,   4.3,  C),

line( 0.0, 0.0, 0.0, 4.0),
            locate (-0.3,   4.3,  D),

line( 0.0, 0.0, 6.0, 4.0),
            line( 6.0, 4.0, 5.8, 3.7),
            line( 6.0, 4.0, 5.7, 3.9),

line( 0.0, 0.0, 0.2, 0.3),
            line( 0.0, 0.0, 0.3, 0.08),

line( 6.0, 0.0, 0.0, 4.0),
            line( 6.0, 0.0, 5.8, 0.3),
            line( 6.0, 0.0, 5.7, 0.08),

line( 0.0, 4.0, 0.2, 3.7),
            line( 0.0, 4.0, 0.3, 3.9),

locate (2.9, 1.9, O)
)}}}
Figure 2. The rectangular ABCD 
and the vectors OA, OB, OC and OD 
 </TD>
 </TR>
</Table>

Answer. &nbsp;The sum of the four vectors that connect the center of a rectangular with its vertices is the zero vector.

<H3>Problem 3</H3>Let &nbsp;ABCD&nbsp; be a rhombus in a plane &nbsp;(Figure 3).
Find the sum of the vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; that connect the center &nbsp;O&nbsp; of the rhombus with its vertices.

<Table>
 <TR>
 <TD>
Solution

A rhombus is the particular case of a parallelogram. &nbsp;It is well known fact that the diagonals&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
of a parallelogram bisect each other, see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A>&nbsp;&nbsp;
in the section Geometry in this site. &nbsp;Hence, the diagonals of a rhombus bisect each other. 
Let us group the sum of the four given vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; as follows:

OA + OB + OC + OD = (OA + OC) + (OB + OD).

The vectors in each parenthesis are lying in one straight line, have the same length and are 
opposite. &nbsp;Hence, the sum of the vectors in each parenthesis is equal to zero. &nbsp;Therefore, 
the sum of the four given vectors is zero.
 </TD>
 <TD>
{{{drawing( 350, 180, -1.0, 9.1, -0.5, 4.5,
 line( 0.0, 0.0, 5.0, 0.0),
 locate (-0.1, -0.1, A),

line( 5.0, 0.0,  8.0, 4.0),
            locate ( 4.9, -0.1,  B),

line( 3.0, 4.0, 8.0, 4.0),
            locate ( 8.1,   4.3,  C),

line( 0.0, 0.0, 3.0, 4.0),
            locate ( 2.7,   4.3,  D),

line( 0.0, 0.0, 8.0, 4.0),
            line( 8.0, 4.0, 7.7, 3.75),
            line( 8.0, 4.0, 7.5, 3.9),

line( 0.0, 0.0, 0.3, 0.25),
            line( 0.0, 0.0, 0.5, 0.1),

line( 5.0, 0.0, 3.0,  4.0),
            line( 5.0, 0.0, 4.92, 0.5),
            line( 5.0, 0.0, 4.6,  0.3),

line( 3.0, 4.0, 3.05, 3.5),
            line( 3.0, 4.0, 3.30, 3.6),

locate (3.8, 1.9, O)
)}}}
Figure 3. The rhombus ABCD 
and the vectors OA, OB, OC and OD 
 </TD>
 </TR>
</Table>

Answer. &nbsp;The sum of the four vectors that connect the center of a rhombus with its vertices is the zero vector.

<H3>Problem 4</H3>Let &nbsp;ABCD&nbsp; be a parallelogram in a plane &nbsp;(Figure 4).
Find the sum of the vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; that connect the center &nbsp;O&nbsp; of the parallelogram with its vertices.

The term "the center of the parallelogram" means the intersection point of the parallelogram diagonals.

<Table>
 <TR>
 <TD>
Solution

It is well known fact that the diagonals of a parallelogram bisect each other, see the lesson&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A>&nbsp; in the section Geometry&nbsp; in this site. 
Let us group the sum of the four given vectors &nbsp;OA, &nbsp;OB, &nbsp;OC &nbsp;and &nbsp;OD&nbsp; as follows:

OA + OB + OC + OD = (OA + OC) + (OB + OD).

The vectors in each parenthesis are lying in one straight line, have the same length and are 
opposite. &nbsp;Hence, the sum of the vectors in each parenthesis is equal to zero. &nbsp;Therefore, 
the sum of the four given vectors is zero.
 </TD>
 <TD>
{{{drawing( 350, 180, -1.0, 9.1, -0.5, 4.5,
 line( 0.0, 0.0, 6.0, 0.0),
 locate (-0.1, -0.1, A),

line( 6.0, 0.0,  8.0, 4.0),
            locate ( 5.9, -0.1,  B),

line( 2.0, 4.0, 8.0, 4.0),
            locate ( 8.1,   4.3,  C),

line( 0.0, 0.0, 2.0, 4.0),
            locate ( 1.7,   4.3,  D),

line( 0.0, 0.0, 8.0, 4.0),
            line( 8.0, 4.0, 7.7, 3.75),
            line( 8.0, 4.0, 7.5, 3.9),

line( 0.0, 0.0, 0.3, 0.25),
            line( 0.0, 0.0, 0.5, 0.1),

line( 6.0, 0.0, 2.0,  4.0),
            line( 6.0, 0.0, 5.7, 0.5),
            line( 6.0, 0.0, 5.4,  0.3),

line( 2.0, 4.0, 2.3, 3.5),
            line( 2.0, 4.0, 2.6, 3.6),

locate (3.8, 1.9, O)
)}}}
Figure 4. The parallelogram ABCD 
and the vectors OA, OB, OC and OD 
 </TD>
 </TR>
</Table>

Answer. &nbsp;The sum of the four vectors that connect the center of a parallelogram with its vertices is the zero vector.

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;and

&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 the vectors that are coherently oriented sides of an unclosed polygon</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;- <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;- Sum of vectors that connect the center of a parallelogram with its vertices&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/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/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.