Question

If \(M\) is the plane through the origin in \(\mathbb{R}^{3}\) with normal \(\mathbf{n}=\left[\begin{array}{l}a \\ b \\ c\end{array}\right],\) show that \(Q_{M}\) has matrix $$ \frac{1}{a^{2}+b^{2}+c^{2}}\left[\begin{array}{ccc} b^{2}+c^{2}-a^{2} & -2 a b & -2 a c \\ -2 a b & a^{2}+c^{2}-b^{2} & -2 b c \\ -2 a c & -2 b c & a^{2}+b^{2}-c^{2} \end{array}\right] $$

Short answer

Derive the reflection matrix using normal vector properties; the matrix confirms reflection properties.

Step-by-step solution

Understand the Problem

We need to find the matrix of the reflection, denoted as \( Q_M \), across a plane \( M \) with the normal vector \( \mathbf{n} = \begin{bmatrix} a \ b \ c \end{bmatrix} \). The equation of the plane through the origin is \( ax + by + cz = 0 \). The key is to use the normal vector properties to derive the reflection matrix.

Derive the Reflection Formula

Reflection of a vector \( \mathbf{v} \) across a plane with normal vector \( \mathbf{n} \) can be described as \( Q_{M} \mathbf{v} = \mathbf{v} - 2 \frac{\mathbf{n} \cdot \mathbf{v}}{\mathbf{n} \cdot \mathbf{n}} \mathbf{n} \). The dot product \( \mathbf{n} \cdot \mathbf{n} = a^2 + b^2 + c^2 \). Using this formula, you can determine how to transform any vector to reflect across the plane.

Determine the General Reflection Matrix

When we express \( \mathbf{v} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) and plug into the reflection formula, the reflection expressed in terms of \( x, y, \, \) and \( z \), results in a matrix operation. Step-by-step calculation can be expanded to show how each entry is formed, using the transformation \( Q_{M}(\mathbf{v}) = \mathbf{v} - 2 \frac{(ax + by + cz)}{a^2 + b^2 + c^2}\begin{bmatrix} a \ b \ c \end{bmatrix} \).

Construct the Matrix

Apply the transformation to construct the matrix structure. Follow the coefficients derived from applying \( Q_{M}(x,y,z) \), leading to the matrix:\[Q_{M} = \frac{1}{a^2 + b^2 + c^2}\begin{bmatrix} b^2 + c^2 - a^2 & -2ab & -2ac \ -2ab & a^2 + c^2 - b^2 & -2bc \ -2ac & -2bc & a^2 + b^2 - c^2 \end{bmatrix} \]

Verify the Matrix Properties

Check the derived matrix by ensuring it's symmetric and if possible, apply it for a practical scenario or specific vector and verify it behaves as expected by reflecting vectors across the plane. Ensuring the matrix properties align with expectations of reflection, such as unitary transformations and conservation of distances.

Key concepts

Plane Through Origin

A plane is a flat, two-dimensional surface that extends infinitely. In this context, the plane is defined in a three-dimensional space, \( \mathbb{R}^{3} \), and it passes through the origin, which is the point where all coordinates equal zero: \( (0, 0, 0) \). The general equation for any plane in \( \mathbb{R}^{3} \) is \( ax + by + cz = d \), but since our plane goes through the origin, the equation simplifies to \( ax + by + cz = 0 \). This simplification is crucial when working out any reflections, as everything is centered around the origin, making calculations consistent and straightforward.
Reflecting a vector across this plane involves projecting the vector onto the plane in such a way that it takes on a mirrored position, relative to where it initially interacted with the plane. The plane's positioning through the origin is fundamental in integrating various mathematical operations, such as reflections.

Normal Vector

The normal vector \( \mathbf{n} = \begin{bmatrix} a \ b \ c \end{bmatrix} \) is vital in defining the orientation of a plane. This vector is perpendicular to every vector that lies in the plane. In geometric terms, this is what determines the 'tilt' or 'slant' of the plane in the three-dimensional space.
In our scenario, the reflection matrix is fundamentally based on this normal vector, as it serves as a mirror line for reflections. The calculation of the reflection involves the dot product of this normal vector with the vector that we want to reflect.
Some key traits of the normal vector include:

• Directly influencing the plane equation \( ax + by + cz = 0 \).
• Used to project vectors onto the plane, a key part of reflection.

In essence, mastering the concept of a normal vector is integral to understanding and applying plane reflections.

Matrix Operations

In mathematics, matrix operations are core to computations in linear algebra. They include addition, subtraction, scalar multiplication, and others. Here, we're particularly concerned with how a matrix can transform vectors through operations like multiplication.
Matrix multiplication, the most relevant operation here, directly ties into how reflections are calculated. When you multiply a reflection matrix by a vector, you adjust the vector's position according to the matrix's specification – in this case, reflecting over a specified plane. A reflection matrix rearranges the vector coordinates by specific multipliers that have been predetermined based on the plane's normal vector.
This principle demonstrates how powerful matrix operations can be for applications in geometry and physics, where spatial transformations are required.

Vector Reflection

Reflecting a vector across a plane involves a series of precise calculations. To reflect a vector \( \mathbf{v} \) using a plane’s normal vector \( \mathbf{n} \), first, you project the vector onto the normal vector through the dot product, then subtract this projection twice from the original vector. This sequence results in a vector mirrored across the plane.
Mathematically, the reflection is represented as \( Q_{M} \mathbf{v} = \mathbf{v} - 2 \frac{\mathbf{n} \cdot \mathbf{v}}{\mathbf{n} \cdot \mathbf{n}} \mathbf{n} \). The term \( \mathbf{n} \cdot \mathbf{v} \) represents the dot product of the normal vector and the vector to be reflected, while \( \mathbf{n} \cdot \mathbf{n} \) normalizes this factor.
This method allows for precise reflection operations, useful in understanding light behavior, building 3D models, and in various engineering applications.

Symmetric Matrix

A symmetric matrix is one where the entries along the diagonal remain unchanged, and for any two indices \( i \) and \( j \), the element in the \( i \)-th row and \( j \)-th column equals the element in the \( j \)-th row and \( i \)-th column, meaning the matrix is mirrored along its diagonal.
In a reflection matrix, symmetry is crucial as it ensures that transformations like reflections maintain certain geometric properties, like distances and angles, consistently across the original and reflected figures.
For example, if you create a reflection matrix \( Q_{M} \), you should notice entries such as \( -2ab \), which appear symmetrically about the diagonal.
Symmetric matrices are fundamental in various fields like physics for defining energy metrics, and in computer graphics for rendering images, thanks to their ability to preserve certain transformation characteristics.