When vectors in the STP are expressed as linear combinations: If p = alphaa+betab, q = gammaa+deltab+epsilon*c, etc., the STP [p q r] can be expanded using multilinearity.

Key result: If p = l1a+m1b+n1c, q = l2a+m2b+n2c, r = l3a+m3b+n3*c, then: [p q r] = |l1 m1 n1; l2 m2 n2; l3 m3 n3| * [a b c].

This is extremely useful: the STP of linear combinations equals the determinant of coefficients times the STP of the base vectors.

Example: If p = a+2b, q = 2a+b+c, r = a+c, then [p q r] = |1 2 0; 2 1 1; 1 0 1| * [a b c] = (1(1-0)-2(2-1)+0)[a b c] = (1-2)[a b c] = -[a b c].