Cartesian Vectors and Quadrilaterals

In summary, the conversation discusses using Cartesian vectors in two-space to prove that the line segments joining midpoints of consecutive sides of a quadrilateral form a parallelogram. The main idea is to use the "parallelogram law" for vector addition, which states that the sum of two vectors is the length of the longer diagonal of the parallelogram formed by those vectors, while the difference of the two vectors is the length of the shorter diagonal.
  • #1
Starcrafty
13
0
I have no clue where to start on this question.
Use Cartesian vectors in two-space to prove that the line segments joining midpoints of the consecutive sides of a quadrilateral form a parallelogram.

Atm all i can deduce from the information is that vectors 2A+2B+2C+2D=0 therefore midpoint vectors A+B+C+D=0 and to prove that it is a parallelogram A+B//C+D and vector A+C//B+D
 
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  • #2
Since this talks about using parallelograms, how about using the "parallelogram law" for vector addition? That is, that for vectors u and v, u+ v is the length of the longer diagonal of the parallelogram having u and v as sides and u- v is the length of the shorter diagonal.
 

Related to Cartesian Vectors and Quadrilaterals

1. What are Cartesian vectors and how are they represented?

Cartesian vectors are mathematical objects that represent magnitude and direction. They are commonly used in physics and engineering to describe the movement of objects in space. In two-dimensional Cartesian coordinates, vectors are represented by a combination of two numbers, known as the vector's components.

2. How are Cartesian vectors added or subtracted?

Cartesian vectors can be added or subtracted by adding or subtracting their components. This is known as vector addition and subtraction. The resulting vector will have a magnitude and direction determined by the combined components of the original vectors.

3. What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude, while a vector has both magnitude and direction. Scalars are represented by a single number, while vectors are represented by a combination of two or more numbers.

4. How are quadrilaterals classified in Cartesian coordinates?

In Cartesian coordinates, quadrilaterals are classified based on the position of their vertices. They can be classified as parallelograms, rectangles, squares, rhombuses, or trapezoids depending on the length and angles of their sides.

5. Can all quadrilaterals be represented by Cartesian vectors?

Yes, all quadrilaterals can be represented by Cartesian vectors. The vertices of a quadrilateral can be represented by points in a Cartesian plane, and the sides of the quadrilateral can be represented by vectors connecting these points. The properties of the quadrilateral can then be determined by analyzing the components of these vectors.

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