Question #41235

This question tests the synthesis of matrix algebra and vector calculus.

The primary trap lies in the direction cosines constraint ($l^2+m^2+n^2=1$), which normalizes the vector $\hat{u}$.

Students who fail to recognize this will be unable to simplify $K^2 = \hat{u}\hat{u}^T - I$ and might attempt a brute-force determinant expansion of a highly complex $3 \times 3$ matrix, leading to algebraic exhaustion.

By mapping the skew-symmetric matrix $K$ to a cross-product operator, the problem elegantly reduces to finding eigenvalues on invariant subspaces (the line spanned by $\hat{u}$ and its orthogonal plane), implicitly utilizing the Lagrange identity for the determinant of the orthogonal block.