# Exploring Parallelogram Diagonals: Vector Algebra

• lolimcool
In summary, exploring parallelogram diagonals using vector algebra involves using the properties of vectors to find the length and direction of the diagonals. This method allows for a more efficient and accurate way to calculate the diagonals compared to traditional geometry methods. Additionally, vector algebra can also be used to determine the area of a parallelogram and solve related problems involving parallelograms.
lolimcool

## Homework Statement

P is the point where the diagonals of the parallelogram abcd intersect one another
let $\alpha = AB$ and $\beta = AD$ and let s and t be scalars such that $AP = sAC$ and $BP = tBD$

use vector algebra to show that
$s(\alpha + \beta) = AP = \alpha + t(\beta - \alpha)$

## The Attempt at a Solution

ok so
s(AC) = AP = AB + t(BD) (s(AC) and AP have the same direction so s(AC) = AP)
AP = AB + t(BD)
AP = AB + BP
AP = AP

is that right?and could someone explain to me the principle of planar independence

lolimcool said:

## Homework Statement

P is the point where the diagonals of the parallelogram abcd intersect one another
let $\alpha = AB$ and $\beta = AD$ and let s and t be scalars such that $AP = sAC$ and $BP = tBD$

use vector algebra to show that
$s(\alpha + \beta) = AP = \alpha + t(\beta - \alpha)$

## The Attempt at a Solution

ok so
The line above is essentially what you are trying to show, so you shouldn't start off by assuming that it is true.

First, you want to show that s($\alpha~+~\beta$) = AP, then show that s($\alpha~+~\beta$) = $\alpha~+t(\beta~-\alpha)$. (Or you can show that AP = $\alpha~+t(\beta~-\alpha)$.)

For the first part, you have
s($\alpha~+~\beta$) = s(AB + AD) = s(AC). From the given information, what is that last expression equal to?
lolimcool said:
s(AC) = AP = AB + t(BD) (s(AC) and AP have the same direction so s(AC) = AP)
AP = AB + t(BD)
AP = AB + BP
AP = AP

is that right?
Generally speaking, you don't want to end with a statement that is obviously true; i.e., that some quantity equals itself.
lolimcool said:
and could someone explain to me the principle of planar independence

Actually, what you have written is a valid "synthetic proof" where you start with what you want to show, then work down to an obviously true statement. The critical part is that each statement be reversible. That way a "standard proof" would start at the bottom and work upward.

That is, as "standard" proof would be:
AP= AP
AP= AB+ BP
AP = AB + t(BD)
AP = s(AC) = AB + t(BD)

By the way, you don't want to say "s(AC) and AP have the same direction so s(AC) = AP". It sounds like you are saying "if two vectors have the same direction then they are equal" which, of course, is not true. You are given that s(AC)= AP.

Mark44 said:
Generally speaking, you don't want to end with a statement that is obviously true; i.e., that some quantity equals itself.

HallsofIvy said:
Actually, what you have written is a valid "synthetic proof" where you start with what you want to show, then work down to an obviously true statement. The critical part is that each statement be reversible.
And this is why I qualified what I said. I didn't want to go into details about the steps being reversible.

## 1. What is a parallelogram?

A parallelogram is a quadrilateral with two pairs of parallel sides. This means that the opposite sides of a parallelogram are equal in length and parallel to each other.

## 2. What are the diagonals of a parallelogram?

The diagonals of a parallelogram are line segments that connect the opposite corners (vertices) of the parallelogram. These line segments intersect at their midpoints.

## 3. How do you calculate the length of a parallelogram's diagonals?

The length of a parallelogram's diagonals can be calculated using the Pythagorean theorem. First, find the length of one side of the parallelogram using the distance formula. Then, use the Pythagorean theorem to find the length of the diagonal.

## 4. How is vector algebra used in exploring parallelogram diagonals?

Vector algebra is used to represent the sides and diagonals of a parallelogram as vectors. By using vector addition and subtraction, we can find the length and direction of the diagonals and determine if they bisect each other.

## 5. What are some real-life applications of exploring parallelogram diagonals using vector algebra?

Exploring parallelogram diagonals using vector algebra has many practical applications, such as in engineering and architecture. It can be used to determine the stability and strength of structures, as well as in navigation and mapping systems. Additionally, the principles of vector algebra can be applied in physics and other sciences to analyze motion and forces.

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