Question

a. Show $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2+\Vert \mathbf{u}-\mathbf{v}{\Vert }^2=2(\Vert \mathbf{u}{\Vert }^2+\Vert \mathbf{v}{\Vert }^2)$ for any vectors $\mathbf{u}$ and $\mathbf{v}$. b. What does this say about parallelograms?

Short answer

The identity verifies the Parallelogram Law for vectors.

Step-by-step solution

Evaluate $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2$ Expression

Start by expanding the expression $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2$. By definition, this norm squared is equal to the dot product of the vector with itself: $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2=(\mathbf{u}+\mathbf{v})\cdot (\mathbf{u}+\mathbf{v})$. Apply the distributive property of the dot product to find: $$\Vert \mathbf{u}+\mathbf{v}{\Vert }^2=\mathbf{u}\cdot \mathbf{u}+2(\mathbf{u}\cdot \mathbf{v})+\mathbf{v}\cdot \mathbf{v}$$ This simplifies to $\Vert \mathbf{u}{\Vert }^2+2(\mathbf{u}\cdot \mathbf{v})+\Vert \mathbf{v}{\Vert }^2$.

Evaluate $\Vert \mathbf{u}-\mathbf{v}{\Vert }^2$ Expression

Now expand the expression $\Vert \mathbf{u}-\mathbf{v}{\Vert }^2$ in a similar way. Using the definition of the dot product, we have: $$\Vert \mathbf{u}-\mathbf{v}{\Vert }^2=(\mathbf{u}-\mathbf{v})\cdot (\mathbf{u}-\mathbf{v})$$ Applying the distributive property yields: $$\Vert \mathbf{u}-\mathbf{v}{\Vert }^2=\mathbf{u}\cdot \mathbf{u}-2(\mathbf{u}\cdot \mathbf{v})+\mathbf{v}\cdot \mathbf{v}$$This simplifies to $\Vert \mathbf{u}{\Vert }^2-2(\mathbf{u}\cdot \mathbf{v})+\Vert \mathbf{v}{\Vert }^2$.

Combine and Simplify

We need to add the two expressions found in steps 1 and 2 together: $$\Vert \mathbf{u}+\mathbf{v}{\Vert }^2+\Vert \mathbf{u}-\mathbf{v}{\Vert }^2=(\Vert \mathbf{u}{\Vert }^2+2(\mathbf{u}\cdot \mathbf{v})+\Vert \mathbf{v}{\Vert }^2)+(\Vert \mathbf{u}{\Vert }^2-2(\mathbf{u}\cdot \mathbf{v})+\Vert \mathbf{v}{\Vert }^2)$$When you simplify this, you can see that the $2(\mathbf{u}\cdot \mathbf{v})$ and $-2(\mathbf{u}\cdot \mathbf{v})$ terms cancel each other out. This leaves us with:$$\Vert \mathbf{u}+\mathbf{v}{\Vert }^2+\Vert \mathbf{u}-\mathbf{v}{\Vert }^2=2\Vert \mathbf{u}{\Vert }^2+2\Vert \mathbf{v}{\Vert }^2$$This equals $2(\Vert \mathbf{u}{\Vert }^2+\Vert \mathbf{v}{\Vert }^2)$.

Interpret the Result

The equality $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2+\Vert \mathbf{u}-\mathbf{v}{\Vert }^2=2(\Vert \mathbf{u}{\Vert }^2+\Vert \mathbf{v}{\Vert }^2)$ applies to any vectors $\mathbf{u}$ and $\mathbf{v}$. This is a geometric interpretation for the diagonal properties of parallelograms. In a parallelogram, the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of all four sides. This is known as the Parallelogram Law.

Key concepts

Vector Norms

Understanding vector norms is a foundational concept in vector mathematics. A vector norm is a quantity that provides a measure of the "size" or "length" of the vector.
Vector norms are crucial for calculating distances between points, determining the magnitude of forces, and more. The most common vector norm is the Euclidean norm, also known as the standard (or L2) norm.

• The Euclidean norm of a vector $\mathbf{u}$ is denoted as $\Vert \mathbf{u}\Vert$.
• It is calculated using the dot product: $\Vert \mathbf{u}\Vert =\sqrt{\mathbf{u}\cdot \mathbf{u}}$.

In the context of the parallelogram law, vector norms are used to evaluate the diagonal and side lengths of parallelograms formed by vectors. Recognizing that these norms are expressed in terms of the dot product makes connections to other vector operations clearer.

Dot Product

The dot product is an essential operation in vector algebra. It establishes a way to multiply two vectors, resulting in a scalar.
This product extends beyond simple multiplication as it reveals angles between vectors and projects vector forces.

• The dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is expressed as: $\mathbf{u}\cdot \mathbf{v}=\sum _{i=1}^n{u}_i{v}_i$ for vectors in $n$-dimensional space.
• This can also be represented using the angle $\theta$ between the two vectors: $\mathbf{u}\cdot \mathbf{v}=\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert \cos \theta$.

The dot product's distributive property is key in simplifying expressions like $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2$ and $\Vert \mathbf{u}-\mathbf{v}{\Vert }^2$. This operation facilitates the demonstration of the Parallelogram Law by simplifying expressions into easily manageable terms.

Geometric Interpretation

Geometric interpretation of vector expressions unlocks deeper understanding of mathematical relationships in physics and engineering.
The Parallelogram Law provides a geometric perspective on how vectors combine.
For instance, when the vectors $\mathbf{u}$ and $\mathbf{v}$ form the sides of a parallelogram, their sums and differences (as in the equations $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2$ and $\Vert \mathbf{u}-\mathbf{v}{\Vert }^2$) align with the properties of the diagonals.

• The sum of the squares of the diagonals' lengths equals twice the sum of the squares of the sides' lengths.
• This equation visually represents how any two non-parallel vectors can stretch across a common origin and skew out to form grid-like patterns.

This law is not only a mathematical curiosity but also serves practical applications, helping to visualize and solve spatial problems, especially in assessing the resultant magnitudes of forces acting in two dimensions.