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Consider the parallelogram with adjacent sides $\mathbf{u}$ and $\mathbf{v}$. a. Show that the diagonals of the parallelogram are $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$ b. Prove that the diagonals have the same length if and only if $\mathbf{u} \cdot \mathbf{v}=0$ c. Show that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.

Consider the parallelogram with adjacent sides $\mathbf{u}$ and $\mathbf{v}$.
a. Show that the diagonals of the parallelogram are $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$
b. Prove that the diagonals have the same length if and only if $\mathbf{u} \cdot \mathbf{v}=0$
c. Show that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.

Key Concepts

Parallelogram Law

The parallelogram law establishes that in any parallelogram, the sum of the squares of the diagonals is equal to the sum of the squares of the sides. This relationship is a fundamental property of vectors that provides a neat balance between the geometry of the shape and its algebraic representation, and it plays a central role in various fields such as physics and engineering.

Vector Addition

This concept involves combining vectors to determine resultant directions and magnitudes. In the context of a parallelogram, the diagonal vectors can be expressed as the sum or difference of the side vectors. This method of vector addition is key to showing that the two diagonals are u+v and u–v.

Dot Product and Orthogonality

The dot product measures the extent to which two vectors point in the same direction, and when the dot product is zero, the vectors are perpendicular. This concept is used to prove that the diagonals of a parallelogram have equal lengths if and only if the sides are orthogonal, directly linking algebraic and geometric properties.

Transcript

00:01 Okay, this question asks us about a parallelogram where we have one of its sides equal to u, and then the adjacent side equal to v.

00:15 So then filling in the rest of the parallelogram, this side would be minus u, and this side would be minus v.

00:27 And it asks us about the diagonals of this shape.

00:31 So for the first diagonal, this one's pretty easy to see because we just go.

00:36 Go u to the right, then v up to make that.

00:40 So this is u plus v.

00:48 And then our other diagonal going here, we see we go u to the right, but then v down.

00:58 So this is u minus v.

01:05 So now that we found this, we have to show that the diagonals have the same length if and only if u .d .v equals zero.

01:17 So the length of the first diagonal is just the magnitude of u plus v or l1 squared is equal to the magnitude of u plus v squared and i wrote it like that because we can write this now in terms of a dot product which we know using foil this evaluates to magnitude of u squared plus the magnitude of v squared plus two times the the dot product of u and v.

02:04 So that's our length of the first diagonal and then we'll do the same thing for the second, which is the magnitude of u minus v, and we'll talk about the square.

02:19 So this gives magnitude of u plus magnitude of v squared each minus two times u.

02:33 And we see here l1 has a plus 2 u .dv and then l2 has a minus 2 u .dv and they're only going to be the same if u .dv equals 0.

02:55 So that's the first part and then we have to prove it going the other way...

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