Question

a. Show that \(\mathbf{x}\) and \(\mathbf{y}\) are orthogonal in \(\mathbb{R}^{n}\) if and only if \(\|\mathbf{x}+\mathbf{y}\|=\|\mathbf{x}-\mathbf{y}\|\) b. Show that \(\mathbf{x}+\mathbf{y}\) and \(\mathbf{x}-\mathbf{y}\) are orthogonal in \(\mathbb{R}^{n}\) if and only if \(\|\mathbf{x}\|=\|\mathbf{y}\|\).

Short answer

a. Orthogonal iff \(\|\mathbf{x} + \mathbf{y}\| = \|\mathbf{x} - \mathbf{y}\|\). b. \(\|\mathbf{x}\| = \|\mathbf{y}\|\) iff orthogonal.

Step-by-step solution

Define Orthogonal Vectors

Two vectors \(\mathbf{x}\) and \(\mathbf{y}\) are orthogonal if their dot product is zero, i.e., \(\mathbf{x} \cdot \mathbf{y} = 0\).

Step 2a: Use the Norm Formula

The norm (or length) of a vector \(\mathbf{z}\) is given by \(\|\mathbf{z}\| = \sqrt{\mathbf{z} \cdot \mathbf{z}}\). For vectors \(\mathbf{x} + \mathbf{y}\) and \(\mathbf{x} - \mathbf{y}\), we can expand using the dot product: \(\|\mathbf{x} + \mathbf{y}\|^2 = (\mathbf{x} + \mathbf{y}) \cdot (\mathbf{x} + \mathbf{y}) = \mathbf{x} \cdot \mathbf{x} + 2(\mathbf{x} \cdot \mathbf{y}) + \mathbf{y} \cdot \mathbf{y}\). Similarly, for \(\mathbf{x} - \mathbf{y}\): \(\|\mathbf{x} - \mathbf{y}\|^2 = \mathbf{x} \cdot \mathbf{x} - 2(\mathbf{x} \cdot \mathbf{y}) + \mathbf{y} \cdot \mathbf{y}\).

Step 3a: Equate the Norms

Since \(\|\mathbf{x} + \mathbf{y}\| = \|\mathbf{x} - \mathbf{y}\|\), we equate their squares: \(\mathbf{x} \cdot \mathbf{x} + 2\mathbf{x} \cdot \mathbf{y} + \mathbf{y} \cdot \mathbf{y} = \mathbf{x} \cdot \mathbf{x} - 2\mathbf{x} \cdot \mathbf{y} + \mathbf{y} \cdot \mathbf{y}\). This simplifies to \(4\mathbf{x} \cdot \mathbf{y} = 0\), which implies \(\mathbf{x} \cdot \mathbf{y} = 0\). Therefore, \(\mathbf{x}\) and \(\mathbf{y}\) are orthogonal.

Conclusion for Part a

\(\mathbf{x}\) and \(\mathbf{y}\) are orthogonal if and only if \(\|\mathbf{x} + \mathbf{y}\| = \|\mathbf{x} - \mathbf{y}\|\).

Step 4b: Orthogonality of Sum and Difference

\(\mathbf{x} + \mathbf{y}\) and \(\mathbf{x} - \mathbf{y}\) are orthogonal if their dot product is zero: \((\mathbf{x} + \mathbf{y}) \cdot (\mathbf{x} - \mathbf{y}) = 0\). Expand this as: \((\mathbf{x} \cdot \mathbf{x}) - (\mathbf{y} \cdot \mathbf{y}) = 0\).

Step 5b: Solve the Dot Product Equation

The equation \((\mathbf{x} \cdot \mathbf{x}) - (\mathbf{y} \cdot \mathbf{y}) = 0\) simplifies to \(\|\mathbf{x}\|^2 = \|\mathbf{y}\|^2\), which implies \(\|\mathbf{x}\| = \|\mathbf{y}\|\) since both are non-negative.

Conclusion for Part b

\(\mathbf{x} + \mathbf{y}\) and \(\mathbf{x} - \mathbf{y}\) are orthogonal if and only if \(\|\mathbf{x}\| = \|\mathbf{y}\|\).

Key concepts

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers, usually vectors, and returns a single number. It's crucial in determining the relationship between vectors. Given two vectors \( \mathbf{x} = (x_1, x_2, \ldots, x_n) \) and \( \mathbf{y} = (y_1, y_2, \ldots, y_n) \), the dot product \( \mathbf{x} \cdot \mathbf{y} \) is calculated as:

• \( \mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2 + \ldots + x_ny_n\)

Assuming vectors are in a real number space \( \mathbb{R}^n \), the dot product is zero when vectors are orthogonal. This product reflects the angle between them: a dot product of zero means that vectors are perpendicular, or at a 90-degree angle. Understanding the dot product is also essential for other vector operations, including projections and measuring vector orthogonality.

Norm of a Vector

The norm of a vector, often called the magnitude or length, measures how long the vector is. For a vector \( \mathbf{z} = (z_1, z_2, \ldots, z_n) \), the norm is calculated using the formula:

• \( \|\mathbf{z}\| = \sqrt{z_1^2 + z_2^2 + \ldots + z_n^2}\)

This is a measure of the vector's length in the vector space. The concept of norm is not only fundamental in vector mathematics but also in functions, spaces, and many applications. In the given exercise, comparing the norms of \( \mathbf{x} + \mathbf{y} \) and \( \mathbf{x} - \mathbf{y} \) gives insights into their properties. If these two expressions have the same norm, certain relationships, like orthogonality, can be deduced about the original vectors \( \mathbf{x} \) and \( \mathbf{y} \). Norms are always non-negative, and a norm of zero implies a zero vector.

Orthogonal Vectors

Orthogonality in vectors refers to perpendicular directions in space. Vectors \( \mathbf{x} \) and \( \mathbf{y} \) are orthogonal if their dot product is zero: \( \mathbf{x} \cdot \mathbf{y} = 0 \). Orthogonal vectors are significant because they represent independent directions in vector space, often simplifying complex vector calculations.

• In the context of the exercise, orthogonal vectors were shown to correspond to equal norms when certain vector operations are performed.
• If \( \|\mathbf{x} + \mathbf{y}\| = \|\mathbf{x} - \mathbf{y}\| \), it implies \( \mathbf{x} \cdot \mathbf{y} = 0 \).
• Similarly, for the vectors \( \mathbf{x} + \mathbf{y} \) and \( \mathbf{x} - \mathbf{y} \) to be orthogonal, \( \|\mathbf{x}\| = \|\mathbf{y}\| \) is necessary.

Orthogonality can be thought of as generalizing the notion of "perpendicular" from geometric space to abstract vector spaces. This concept is widely used in fields like computer graphics, physics, and machine learning due to its simplicity and ability to convey independence between elements.