Geometric Horizontal Parallelogram

Geometric Vertical Parallelogram

That's our parallelogram.

This is a parallelogram

Find the perimeter of the parallelogram. We have a parallelogram right over here, opposite sides are parallel.

The parallelogram generated by v2 and v1.

Hey, if we have this parallelogram ABCD,

A parallelogram, the diagonals bisect each other

We know that for any parallelogram, and a rhombus is a parallelogram, that the diagonals bisect each other

The opposite sides of a parallelogram are parallel.

Then the parallelogram would look something like this.

So, we have a parallelogram right over here

In a parallelogram, the opposite sides are parallel

So the area of your parallelogram squared is equal to the determinant of the matrix whose column vectors construct that parallelogram.

That's what the area of a parallelogram would be.

That's what the area of our parallelogram squared is.

You give it a push, this becomes a parallelogram.

And what's the height of this parallelogram going to be?

Given that we wanna prove that this is a parallelogram

And also in a parallelogram, opposite sides have the same length.

A parallelogram is a quadrilateral formed from two sets of parallel lines.

Because it's a parallelogram, we know that the diagonals bisect each other

First of all, a rhombus is a special case of a parallelogram

In general, if I have just any parallelogram this is kind of a tilted one, but if I just have any parallelogram, let me just draw any parallelogram right there the area is just equal to the base so that could be the base times the height.

The area of our parallelogram is equal to the base times the height.

And if opposite sides have the same length then you have a parallelogram

So the shaded portion is the whole square minus the area of the parallelogram.

So remember, a rhombus is just a parallelogram, where all four sides are equal.

In fact, if all four sides are equal, it has to be a parallelogram.

What I mean by that is, imagine that these two guys are position vectors that are specifying points on a parallelogram, and then of course the or not of course but, the origin is also another point in the parallelogram, so what will be the last point on the parallelogram?

A square is one type of rectangle, and a rectangle is one type of parallelogram.

There's actually the area of the parallelogram created by the column vectors of this matrix.

So that's why the area of a parallelogram is just the base times the height.

We've shown if you have a parallelogram, opposite sides are, opposite sides have the same length

Hey, the opposite sides of this quadrilateral must be parallel or that ABCD is a parallelogram

What I wanna do this video is prove that the opposite angles of a parallelogram are congruent

A parallelogram, we already have two sides of it, so the other two sides have to be parallel.

So the base squared we already saw, the base of our parallelogram is the length of vector v1.

What we're going to prove in this video is a couple of fairly straight forward parallelogram related proofs

Now, what we're going to concern ourselves with in this video is the parallelogram generated by these two guys.

What we're going to concern ourselves with specifically is the area of the parallelogram generated by v1 and v2.

There's other interesting things we know about the diagonals of a parallelogram, which we know all rhombi are parallelograms

Which essentially just says that if we had a field the size and shape of this parallelogram, and we wanted to make a fence around the field that was the shape of this parallelogram, how long would that fence have to be?

So there is actually a formula for the area of a parallelogram, it's actually just the base times the height.

If you switched v1 and v2, you're still spanning the same parallelogram, you just might get the negative of the determinant.