Question

Diagonals of a parallelogram 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.

Short answer

Question: Prove that in a parallelogram with adjacent sides given by vectors $\mathbf{u}$ and $\mathbf{v}$, the diagonals have the same length if and only if $\mathbf{u}\cdot \mathbf{v}=0$. Answer: The diagonals of a parallelogram with adjacent sides $\mathbf{u}$ and $\mathbf{v}$ are given by $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$. These diagonals have the same length if and only if $\mathbf{u}\cdot \mathbf{v}=0$.

Step-by-step solution

Define the vertices of the parallelogram

Let's denote the vertices of parallelogram as $A$, $B$, $C$, and $D$ such that $A$ and $D$ connect the sides $\mathbf{u}$ and $\mathbf{v}$. We can define the position vectors of these vertices accordingly: $A:\mathbf{0}$ $B:\mathbf{u}$ $C:\mathbf{u}+\mathbf{v}$ $D:\mathbf{v}$

Find the diagonals of the parallelogram

Diagonals are lines connecting opposite vertices of the parallelogram. So we have two diagonals, $AC$ and $BD$. To find the vector representing these diagonals, we subtract the position vectors of the endpoints: $\overrightarrow{AC}=\mathbf{c}-\mathbf{a}=\mathbf{u}+\mathbf{v}-\mathbf{0}=\mathbf{u}+\mathbf{v}$ $\overrightarrow{BD}=\mathbf{d}-\mathbf{b}=\mathbf{v}-\mathbf{u}=\mathbf{u}-\mathbf{v}$

Prove that diagonals have same length if and only if $\mathbf{u}\cdot \mathbf{v}=0$

The length of a vector $\mathbf{w}$ is given by $||\mathbf{w}||=\sqrt{\mathbf{w}\cdot \mathbf{w}}$. The diagonals will have the same length if $||\mathbf{u}+\mathbf{v}||=||\mathbf{u}-\mathbf{v}||$. We can square both sides to get rid of the square root: $||\mathbf{u}+\mathbf{v}|{|}^2=||\mathbf{u}-\mathbf{v}|{|}^2$. This can be expanded out to yield: $(\mathbf{u}+\mathbf{v})\cdot (\mathbf{u}+\mathbf{v})=(\mathbf{u}-\mathbf{v})\cdot (\mathbf{u}-\mathbf{v})$ Expanding the dot products and simplifying, we get: $\mathbf{u}\cdot \mathbf{u}+\mathbf{v}\cdot \mathbf{v}+2\mathbf{u}\cdot \mathbf{v}=\mathbf{u}\cdot \mathbf{u}+\mathbf{v}\cdot \mathbf{v}-2\mathbf{u}\cdot \mathbf{v}$ Now, we subtract $\mathbf{u}\cdot \mathbf{u}+\mathbf{v}\cdot \mathbf{v}$ from both sides and get: $4\mathbf{u}\cdot \mathbf{v}=0$ This means that the diagonals have the same length if and only if $\mathbf{u}\cdot \mathbf{v}=0$.

Show that the sum of the squares of the diagonals equals the sum of squares of the sides

Let's compute the sum of the squares of the lengths of the diagonals: $||\mathbf{u}+\mathbf{v}|{|}^2+||\mathbf{u}-\mathbf{v}|{|}^2=(\mathbf{u}+\mathbf{v})\cdot (\mathbf{u}+\mathbf{v})+(\mathbf{u}-\mathbf{v})\cdot (\mathbf{u}-\mathbf{v})$ When we expand and simplify this expression, we get: $2(\mathbf{u}\cdot \mathbf{u}+\mathbf{v}\cdot \mathbf{v})=2(||\mathbf{u}|{|}^2+||\mathbf{v}|{|}^2)$ Thus, the sum of the squares of the lengths of the diagonals equals twice the sum of the squares of the lengths of the sides.

Key concepts

Vector Operations

Understanding vector operations is critical for solving a variety of problems in mathematics and physics, particularly when dealing with geometric objects like parallelograms. A vector in this context can be viewed as a directed line segment characterized by both magnitude and direction, and is commonly represented by boldface letters or arrows over the letters.

In the realm of vector operations, addition and subtraction play fundamental roles. When you add vectors, you're essentially combining their magnitudes and directions to form a new vector. Conversely, subtracting one vector from another effectively reveals the vector that spans from the head of the subtracted vector to the head of the one from which it is subtracted. This is particularly relevant to the diagonals of a parallelogram, as these can be expressed as the sum or difference of the parallelogram's adjacent side vectors.

Dot Product

The dot product, also known as the scalar product, is an operation that combines two vectors to produce a scalar, which is a single number. It is determined through the multiplication of the vectors' magnitudes and the cosine of the angle between them, symbolized as $\mathbf{u}\cdot \mathbf{v}$.

This operation is incredibly useful when we want to find the angle between vectors or determine if the vectors are perpendicular. If the dot product of two vectors is zero, it indicates that the vectors are orthogonal, meaning they meet at a right angle. This concept is pivotal in proving that the diagonals of a parallelogram are of identical length only if they are perpendicular to each other, as stated in the problem's part b.

Parallelogram Properties

A parallelogram is a four-sided figure where opposite sides are parallel and equal in length. This geometric shape has several interesting properties that are regularly utilized in vector calculus and geometry.

For instance, the opposite angles of a parallelogram are congruent, and the diagonals bisect each other. This implies that each diagonal divides the parallelogram into two congruent triangles. Furthermore, the diagonals themselves can be represented as vectors, making it straightforward to apply vector operations to analyze their properties such as length equality and relations to the sides of the parallelogram.

Vector Length

The length of a vector, also referred to as magnitude, is a measure of how long the vector is. You can calculate a vector’s length using the Pythagorean theorem in two or three dimensions. In general terms, the length of vector $\mathbf{w}$ is found using the formula $||\mathbf{w}||=\sqrt{\mathbf{w}\cdot \mathbf{w}}$.

This formula plays a critical role in measuring not just single vector lengths but also in comparing the magnitudes of different vectors, such as the diagonals of a parallelogram. The Pythagorean-like relationship presented in part c of our problem ties together the lengths of the sides and diagonals of the parallelogram, showing the interplay between the geometric and algebraic representations of vectors.