= Solution
For the <quadratic form of a squared cross product>, express the two scalar terms as <quadratic forms> in $\mathbf a$:
$$
\boxed{M=|\mathbf b|^2I-\mathbf b\mathbf b^T,\qquad
M_{ij}=|\mathbf b|^2\delta_{ij}-b_ib_j.}
$$
Then $\mathbf a^TM\mathbf a=|\mathbf b|^2|\mathbf a|^2-(\mathbf a\cdot\mathbf b)^2$. In particular, for $\mathbf b=(b_1,b_2,0)$,
$$
M=\begin{pmatrix}b_2^2&-b_1b_2&0\\-b_1b_2&b_1^2&0\\0&0&b_1^2+b_2^2\end{pmatrix},\qquad\boxed{\det M=0.}
$$
The cancellation is structural: $M\mathbf b=0$, while its action on every vector perpendicular to a nonzero $\mathbf b$ is multiplication by $|\mathbf b|^2$. Its <eigenvalues> are therefore $0,|\mathbf b|^2,|\mathbf b|^2$; for $\mathbf b=0$ the whole matrix is zero.
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