A Direct Proof That a 3 × 3 Skew-Symmetric Matrix Sends Its Defining Vector to Zero
Let v = (p, q, r) T be a vector in ℝ³. From its three components, form the 3 × 3 skew-symmetric matrix K = ( 0 −r q r 0 −p −q p 0 ) . The entries reflected across the main diagonal have opposite signs, while every diagonal entry is zero. Therefore, K T = −K. Multiplication by the vector Matrix-vector multiplication can be written as a linear combination of the columns of K. The first column is multiplied by p, the second by q, and the third by r: K v = p ( 0 r −q ) + q ( −r 0 p ) + r ( q −p 0 ) . Distributing p, q and r gives K v = ( 0 · p + (−r) · q + q · r r · p + 0 · q + (−p) · r −q · p + p · q + 0 · r ) . Combining the entries in each row produces ...