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

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

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

Problem 1

Let ABCD be a square in a plane (Figure 1).
Find the sum of the vectors OA, OB, OC and OD that that connect the center O of the square with its vertices.

Solution
A square is the particular case of a parallelogram. It is well known fact that the diagonals
of a parallelogram bisect each other, see the lesson Properties of diagonals of parallelograms
in the section Geometry in this site. Hence, the diagonals of a square bisect each other.
Let us group the sum of the four given vectors OA, OB, OC and OD 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. Hence, the sum of the vectors in each parenthesis is equal to zero. Therefore,
the sum of the four given vectors is zero.

Figure 1. The square ABCD
and the vectors OA, OB, OC and OD

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

Problem 2

Let ABCD be a rectangular in a plane (Figure 2).
Find the sum of the vectors OA, OB, OC and OD that connect the center O of the rectangular with its vertices.

Solution
A rectangular is the particular case of a parallelogram. It is well known fact that the diagonals
of a parallelogram bisect each other, see the lesson Properties of diagonals of parallelograms
in the section Geometry in this site. Hence, the diagonals of a rectangular bisect each other.
Let us group the sum of the four given vectors OA, OB, OC and OD 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. Hence, the sum of the vectors in each parenthesis is equal to zero. Therefore,
the sum of the four given vectors is zero.

Figure 2. The rectangular ABCD
and the vectors OA, OB, OC and OD

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

Problem 3

Let ABCD be a rhombus in a plane (Figure 3).
Find the sum of the vectors OA, OB, OC and OD that connect the center O of the rhombus with its vertices.

Solution
A rhombus is the particular case of a parallelogram. It is well known fact that the diagonals
of a parallelogram bisect each other, see the lesson Properties of diagonals of parallelograms
in the section Geometry in this site. Hence, the diagonals of a rhombus bisect each other.
Let us group the sum of the four given vectors OA, OB, OC and OD 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. Hence, the sum of the vectors in each parenthesis is equal to zero. Therefore,
the sum of the four given vectors is zero.

Figure 3. The rhombus ABCD
and the vectors OA, OB, OC and OD

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

Problem 4

Let ABCD be a parallelogram in a plane (Figure 4).
Find the sum of the vectors OA, OB, OC and OD that connect the center O of the parallelogram with its vertices.

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

Solution
It is well known fact that the diagonals of a parallelogram bisect each other, see the lesson
Properties of diagonals of parallelograms in the section Geometry in this site.
Let us group the sum of the four given vectors OA, OB, OC and OD 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. Hence, the sum of the vectors in each parenthesis is equal to zero. Therefore,
the sum of the four given vectors is zero.

Figure 4. The parallelogram ABCD
and the vectors OA, OB, OC and OD

Answer. 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
    - Sum of the vectors that are coherently oriented sides of a convex closed polygon and
    - Sum of the vectors that are coherently oriented sides of an unclosed polygon
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
    - Sum of vectors that connect the center of a parallelogram with its vertices                              (this lesson)
    - 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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