The STP [a b c] = a.(b x c) is computed as the determinant |a1 a2 a3; b1 b2 b3; c1 c2 c3|. This determinant can be evaluated by expansion along any row or column, cofactor methods, or row operations.

Fundamental properties: (1) Cyclic permutation preserves the value: [a b c]=[b c a]=[c a b]. (2) Transposition reverses the sign: [a b c]=-[b a c]. (3) If any two vectors are equal or proportional, the STP is zero. (4) The STP is multilinear: [pa+qd, b, c]=p[a b c]+q[d b c].

Dot-cross interchange: a.(b x c) = (a x b).c. This allows flexibility in computation.

For vectors expressed as linear combinations of a basis {a, b, c}: if p=l1a+m1b+n1c, q=l2a+m2b+n2c, r=l3a+m3b+n3c, then [p q r]=det(coefficient matrix)*[a b c]. This is extremely powerful for evaluating STPs without expanding each vector.

Important special results: [a+b, b+c, c+a]=2[a b c]. [a-b, b-c, c-a]=0 (since the three vectors sum to zero). [ka, kb, kc]=$k^3$[a b c].

The STP can also be computed as [a b c]^2 = det(Gram matrix), where the Gram matrix has entries $a_i$.$a_j$. This connects the STP to the inner product structure.

Computational tip: when evaluating determinants, look for zeros in the matrix to simplify expansion. Row or column operations (which add multiples of one row to another) preserve the determinant value.