Geometric Proofs Using Vectors

Now that we understand the relationship between the scalar product of two vectors, the angle between them, and their magnitudes, we can prove some interesting geometric properties of quadrilaterals.

We will use the scalar product to show that:

the diagonals of a parallelogram meet at right angles if and only if it is a rhombus
the midpoints of the sides of a quadrilateral join to form a parallelogram
the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides

There are of course many other geometric properties that can be proven with the scalar product, but this article will give you an understanding of how we can approach these.

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