Key properties:

1. Cyclic permutation preserves value: [a b c] = [b c a] = [c a b].
2. Transposition (swap of two vectors) reverses sign: [a b c] = -[b a c] = -[a c b] = -[c b a].
3. If any two vectors are equal: [a a c] = 0 (determinant with two equal rows).
4. If any two vectors are parallel: [a b c] = 0 (one row is a scalar multiple of another).
5. Linearity: [ka b c] = k[a b c]. More generally, [pa+qd, b, c] = p[a b c]+q[d b c].
6. [a b c] = 0 if and only if a, b, c are coplanar.
7. For unit vectors i, j, k: [i j k] = 1.