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December 29th, 2018, 02:16 PM   #1
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Application of vectorial knowledge

I was ask to prove some theorems using my knowledge on vector; please can anyone help me on how to go about this
1. Prove that the diagonals of a rhombus meet at a right angle
2. Prove that the diagonal of a square makes an angle of 45 degrees with all sides
3. Prove the diagonals of a rectangle are equal
4. Prove that the adjacent side of a kite are equal
How am I supposed to prove all this using vectors, please can anyone help me, am seriously in need of how to do it...?

Last edited by skipjack; December 31st, 2018 at 12:13 AM.
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December 29th, 2018, 02:25 PM   #2
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I don't know what the requirement (knowledge on vector) means. However:
1) Symmetry
2) diagonals are angle bisectors
3) symmetry or Pythagorean theorem twice
4) definition of kite?
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December 29th, 2018, 02:59 PM   #3
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Quote:
Originally Posted by mathman View Post
I don't know what the requirement (knowledge on vector) means. However:
4) definition of kite?
a kite is a quadrilateral with pairs of adjacent sides equal in length.

think of a typical kite.
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December 29th, 2018, 08:50 PM   #4
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Please show me a sample with the previous ones...
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December 30th, 2018, 03:18 PM   #5
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Quote:
Originally Posted by romsek View Post
a kite is a quadrilateral with pairs of adjacent sides equal in length.

think of a typical kite.
That's what I thought. So 4) is true by definition.
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December 30th, 2018, 04:18 PM   #6
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Here's how you might get started for problem 1:

A rhombus is a parallelogram, all of whose sides are equal in length. If we want to think of this in terms of vectors, consider two vectors $\vec{v_1}$ and $\vec{v_2}$. If you draw a picture of this, you will see that $\vec{v_1} + \vec{v_2}$ will be the diagonal of a parallelogram formed by these two vectors. The other diagonal will be $\vec{v_1} - \vec{v_2}$ (or equivalently, $\vec{v_2} - \vec{v_1}$).

This is a rhombus if $|| \vec{v_1}|| = || \vec{v_2}||$.

So how can you tell if the two diagonals ($\vec{v_1} + \vec{v_2}$) and ($\vec{v_1} - \vec{v_2}$) are perpendicular to each other? (Hint: think dot product.)
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December 30th, 2018, 04:18 PM   #7
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Originally Posted by Harmeed View Post
Please show me a sample with the previous ones...
Tell you what.

You have been given much information about 4). Can you set up a proof for this one?

-Dan
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