Prove Rhombus Diagonals Perpendicular: Vector Homework

In summary, to prove that the diagonals of a parallelogram are perpendicular, it must be shown that (a+b) (dot) (a-b) = 0, which is true if and only if the magnitudes of a and b are equal. This can be proven by multiplying out the equation and simplifying to a² = b². Therefore, the diagonals of a parallelogram are perpendicular if and only if the parallelogram is a rhombus.
  • #1
prace
102
0

Homework Statement


Prove that the diagonals of a paraelogram are perpendicular iff the parallelogram is a rhombus.

Homework Equations



a (dot) b = 0

The Attempt at a Solution


This is how I started:

By definition, a rhombus is a quadrilateral with all sides equal in length. So this means that if I have two vectors, a and b that form the corner of a rhombus, then that means that the magintude of a and b are equal. By inspection of a diagram of this vector problem, I found that (a+b) (dot) (a-b) = 0 iff the magintude of a and b are equal.

This is great, however, it will not fly because I cannot just say "by inspection of the diagram" right. How can I put this in words that will make my proof make sense?

Thanks
 
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  • #2
The diagonals of the parallelogram are precisely a+b and a-b. if you are talking about proving your equation above, multiply it out, keeping in mind:

[tex] (a+b)\cdot (a-b) = a\cdot a - b\cdot b + b\cdot a - a\cdot b[/tex]

edit: fixed my mistake
 
Last edited:
  • #3
slearch said:
The diagonals of the parallelogram are precisely a+b and a-b. if you are talking about proving your equation above, multiply it out, keeping in mind:

[tex] (a+b)\cdot (a-b) = a\cdot a + b\cdot b + b\cdot a - a\cdot b[/tex]

AWESOME! Thank you so much for you help. That was a lot easier than I thought. So, after multiplying it out, I came up with a²-b²=0. So this is true iff a² = b². Thanks for your help!
 

Related to Prove Rhombus Diagonals Perpendicular: Vector Homework

1. What is a rhombus?

A rhombus is a type of parallelogram with four equal sides. It has two pairs of parallel sides and opposite angles are equal. It can also be referred to as a diamond shape.

2. How do you prove that the diagonals of a rhombus are perpendicular?

To prove that the diagonals of a rhombus are perpendicular, we can use the fact that opposite sides of a rhombus are parallel to each other. We can also use the properties of vectors, specifically the dot product. If the dot product of two vectors is equal to 0, it means they are perpendicular. Therefore, if we can show that the dot product of the diagonals is equal to 0, we can prove that they are perpendicular.

3. What are vectors?

Vectors are mathematical quantities that have both magnitude (size) and direction. They are typically represented by an arrow pointing in the direction of the vector with a length proportional to its magnitude. In geometry, vectors can be used to represent lines, segments, and other geometric figures.

4. Why is it important to prove that the diagonals of a rhombus are perpendicular?

Proving that the diagonals of a rhombus are perpendicular is important because it is one of the defining properties of a rhombus. It helps us to identify and distinguish a rhombus from other types of quadrilaterals. Additionally, this property can be used to solve problems and prove other geometric theorems.

5. How can I apply this knowledge to real-life situations?

The concept of proving that the diagonals of a rhombus are perpendicular can be applied to real-life situations in many ways. For example, it can be used in engineering and architecture to ensure that structures are built with correct angles and dimensions. It can also be used in navigation, as directions and bearings can be represented using vectors. Additionally, understanding this concept can help with problem-solving and critical thinking skills in various fields.

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