Final answer:
The reflection of a vector in three dimensions is a transformation that produces the mirror image of the original vector across a plane. This is represented mathematically by applying a reflection matrix to the vector. The law of reflection states that the angles of incidence and reflection are equal.
Step-by-step explanation:
Reflection of a Vector in ℝ³
The reflection of a vector in three-dimensional space, denoted as t: ℝ³ → ℝ³, involves changing the direction of the vector to its mirror image with respect to a reflecting plane while maintaining its magnitude. This operation can be described by a reflection matrix depending on the plane of reflection. To find the reflected vector ¨R, you will apply this matrix to the original vector.
The law of reflection states that the angle of incidence (θi) is equal to the angle of reflection (θr), which is written as θi = θr. This law is also applicable to the reflection of vectors in physics and is fundamental in geometrical optics.
To analyze the reflection further, one would need the specific plane of reflection or the normal vector to this plane. Reflections are linear transformations and can be represented using linear algebra techniques. For any vector operation, including reflections, resultant vectors can be computed using vector addition or subtraction, and their magnitudes and directions can be identified using trigonometric identities and vector norms.