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Proving a geometric maxim

  • #1
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Homework Statement



Using vectors, the dot product, and the cross product, prove that the sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of two adjacent sides of the parallelogram.

Homework Equations



[tex]|A·B|=|A||B|cosθ[/tex]

[tex]|AxB|=|A||B|sinθ[/tex]

The Attempt at a Solution



I used the Pythagorean theorem to solve it easily. But I don't know how to solve it using vectors. Is the problem expecting me to draw a parallelogram using arrows? I could create the same triangle I made to solve it using the P-theorem, but that would just be making a triangle. I need to make a parallelogram. It seems like the best the dot and cross product could do is tell me the angle of the triangle. I don't see how it's going to give me the square of anything.
 

Answers and Replies

  • #2
SteamKing
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Let's say two of the sides of a typical parallelogram are formed by the vectors A and B.
Using vector algebra, how would you express the formula for the two diagonals of the parallelogram in terms of A and B? Once you had these expressions, can you prove the original problem about the lengths of the diagonals?

You mentioned the Pythagorean Theorem (P.T.) How does the P.T. relate to finding the magnitude of a vector?
 
  • #3
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Let's say two of the sides of a typical parallelogram are formed by the vectors A and B.
Using vector algebra, how would you express the formula for the two diagonals of the parallelogram in terms of A and B? Once you had these expressions, can you prove the original problem about the lengths of the diagonals?

You mentioned the Pythagorean Theorem (P.T.) How does the P.T. relate to finding the magnitude of a vector?
Ok, if I had the bottom side of a rectangle (A) and the right side of a rectangle (B), then I could find one diagonal with A+B. And then to find the other diagonal, I could do A+(-B). I can get a length of the diagonals if I had the length of A and B. But I'd be using the PT.

To use the PT, I had to come up with my own numbers for the vector magnitudes. I just used 4 and 3, since that makes the diagonal 5.
 
  • #4
SteamKing
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Well, I suppose this is for a math class, and math teachers like for you to use equations rather than plugging in arbitrary numbers. After all, math isn't supposed to be an experimental science.
 
  • #5
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I can't imagine why you are asked to use vectors' crossproduct in a 2d world. You only need to use the sum of vectors to tackle this one.
 
Last edited:
  • #6
HallsofIvy
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Given any parallelogram we can set up a coordinate system so that one vertex is at the origin and another is on the x- axis. That is, one vertex is at (0, 0), another at (a, 0), a third at (b, c) and the fourth at (a+ b, c). What are the vectors giving the diagonals and adjacent sides?
 
  • #7
vela
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Ok, if I had the bottom side of a rectangle (A) and the right side of a rectangle (B), then I could find one diagonal with A+B. And then to find the other diagonal, I could do A+(-B). I can get a length of the diagonals if I had the length of A and B. But I'd be using the PT.
Use ##\| \vec{X} \| = \sqrt{\vec{X}\cdot\vec{X}}## on A+B and A-B.
 

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