Reflections calculator

This page provides two calculators related to calculating reflections in a the real Euclidean vector space \( \mathbb{R}^n \) with the standard inner product

The reflection

This calculator computes \( s_\alpha(v) \), the reflection of a vector \( v\) through the hyperplane perpendicular to \(\alpha\).

Input

Enter the vector \(\alpha\) as a comma seperated list with parenthesis, e.g:

(-3, 0, 1, 4.5)

Enter the vector \(v\) as above:

Result

Your answer will display here.

Vectors generated by reflections

The calculator below allows you to enter a list of vectors \(v_1, v_2,\ldots,v_p \) and will return a list of vectors that is the result of applying \(s_{v_i}(v_j) \) for every pair of vectors.

Input

Put your input in the box below, formatting vectors as above and separating each vector with a line break like this:

          (-3, 0, 1, 4.5)
          (1, 0, -1, 2)
          (1.5, 1, 0, 0)

Result

Your answer will display here.