Math Woods | Free Mathematics Problems and Concepts

Solution

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Set $A=\left(a_{i j}\right), u=\left(u_1, u_2, u_3\right)^{\top}$ . If we use the orthogonality condition $(A u, u)=0$ with $u_i=\delta_{i k}$ we get $a_{k k}=0$ . If we use (1) with $u_i=\delta_{i k}+\delta_{i m}$ we get
$a_{k k}+a_{k m}+a_{m k}+a_{m m}=0$ and hence $a_{k m}=-a_{m k}$ .
Set $v_1=-a_{23}, v_2=a_{13}, v_3=-a_{12}$ . Then
$A u=\left(v_2 u_3-v_3 u_2, v_3 u_1-v_1 u_3, v_1 u_2-v_2 u_1\right)^{\top}=v \times u .$