행렬식

| a_11 | = a_11

| {{a_11, a_12}, {a_21, a_22}} | = a_11a_22 - a_12a_21 = a_11a_22 - a_21a_12

3차원 벡타의 외적

U×V = (u_1, u_2, u_3) × (v_1, v_2, v_3)

e_1 = (1, 0, 0)

e_2 = (0, 1, 0)

e_3 = (0, 0, 1)

U×V = | {{e_1, e_2, e_3}, {u_1, u_2, u_3}, {v_1, v_2, v_3}} | = e_1 | {{u_2, u_3}, {v_2, v_3}} | -e_2 | {{u_1, u_3}, {v_1, v_3}} | +e_3 | {{u_1, u_2}, {v_1, v_2}} |

= (u_2v_3 - u_3v_2) e_1 + (u_3v_1 - u_1v_3) e_2 + (u_1v_2 - u_2v_1) e_3

= (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)

벡타외적의 기본성질

e_1×e_2 = e_3

e_2×e_3 = e_1

e_3×e_1 = e_2

e_2×e_1 = -e_3

e_3×e_2 = -e_1

e_1×e_3 = -e_2

(k u) ×v = k (u×v)

u× (k v) = k (u×v)

u× (v + w) = u×v + u×w

(u + v) ×w = u×w + v×w

u×v = -v×u

u× (v×w) ≠ (u×v) ×w

u× (v×w) = <u, w>v - <u, v>w

<u, v×w> = | {{u_1, u_2, u_3}, {v_1, v_2, v_3}, {w_1, w_2, w_3}} |