find vector reflected across another vector

Question

I apologize that this may sound like a very basic question, but I can't find any clear answers in my search. I have a vector $\vec{v}$ that I want to reflect across a vector $\vec{n}$. The dot product between the two vectors will be positive, in other words, they will both point outward, and the angle between them will be less than 90 degrees. How do I find the vector that results from reflecting $\vec{v}$ across $\vec{n}$?

Answer 1

Approach: Restrict your attention to $\vec{n}$ which lie on any line that passes through the origin. Then the question can be raised "how do I reflect $\vec{v}$ across any line passing through the origin?", and the answer can be given in terms of $\vec{n}$, as you will find here. This is the essential answer, and reflecting about a line not passing through the origin simply requires a not-so-hard-to-formulate "shift".

Answer 2

I would suggest this means that the component of $\vec v$ parallel to $\vec n$ stays the same and the component orthogonal to $\vec n$ changes sign. You can easily use dot product to give a formula.

Answer 3

If you make $\vec{n}$ a unit vector, you should be able to find an orthonormal basis $\{\vec{v}_1,\vec{v}_2,\vec{n}\}$ (note $\vec{v}_1$ and $\vec{v}_2$ will span the plane to which $\vec{n}$ is the normal vector). Then reflection across the vector $\vec{n}$ will yield:$$c_1\vec{v}_1+c_2\vec{v}_2+c_3\vec{n}\mapsto -c_1\vec{v}_1-c_2\vec{v}_2+c_3\vec{n}.$$It would be easy to create the matrix that represents this linear transformation in this basis.

Answer 4

This occurs in the Householder transformation.

To quote wikipedia, "In linear algebra, a Householder transformation (also known as Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and in the first step of the QR algorithm. The Householder transformation was introduced in 1958 by Alston Scott Householder.[1]".

(The reference is Householder, A. S. (1958). "Unitary Triangularization of a Nonsymmetric Matrix". Journal of the ACM 5 (4): 339–342. doi:10.1145/320941.320947. MR 0111128.)

To quote further, "The reflection hyperplane can be defined by a unit vector $v$ (a vector with length $1$) which is orthogonal to the hyperplane. The reflection of a point x about this hyperplane is: $x-2<v,x> =x-2v(v^Hx) $ where $v$ is given as a column unit vector with Hermitian transpose $v^H$. This is a linear transformation given by the Householder matrix: $P=I-2v v^h$, where $I$ is the identity matrix."