向量积：
设 a = A i + B j + C k , b = D i + E j + F k , a × b = ( A i + B j + C k ) ( D i + E j + F k ) = A D i × i + A E i × j + A F i × k + B D j × i + B E j × j + B F j × k + C D k × i + C E k × j + C F k × k = A E k − A F j − B D k + B F i + C D j − C E i = ∣ A B C D E F i j k ∣ 设 a = Ai + Bj + Ck , b = Di + Ej + Fk, a \times b = (Ai + Bj + Ck) (Di + Ej + Fk) = \\ AD i\times i + AEi \times j + AF i \times k + \\ BD j \times i+ BE j\times j + BF j\times k + \\ CD k\times i+ CE k\times j + CF k\times k = \\ AE k - AF j - BD k + BF i + CD j - CE i = \begin{vmatrix} A & B & C \\ D & E & F\\i&j&k \end{vmatrix} 设a=Ai+Bj+Ck,b=Di+Ej+Fk,a×b=(Ai+Bj+Ck)(Di+Ej+Fk)=ADi×i+AEi×j+AFi×k+BDj×i+BEj×j+BFj×k+CDk×i+CEk×j+CFk×k=AEk−AFj−BDk+BFi+CDj−CEi= ADiBEjCFk
尤其要注意的是向量积中的符号！
混合积：
已知a,b,c三个向量，先做a和b的向量积 a × b a \times b a×b，与c再做数量积 [ a × b ] ⋅ c [a\times b] \cdot c [a×b]⋅c，结果即为a,b,c的混合积。设 c = H i + I j + J k c = Hi + I j + J k c=Hi+Ij+Jk
[ a × b ] ⋅ c = [ ∣ A B D E ∣ k + ∣ C A F D ∣ j + ∣ B C E F ∣ i ] ⋅ c = ∣ A B D E ∣ H + ∣ C A F D ∣ I + ∣ B C E F ∣ J = ∣ A B C D E F H I J ∣ [a\times b] \cdot c = [ \begin {vmatrix} A & B\\D & E \end {vmatrix}k + \begin {vmatrix} C & A\\F& D \end {vmatrix}j + \begin {vmatrix} B & C\\E& F \end {vmatrix}i ] \cdot c =\\ \begin {vmatrix} A & B\\D & E \end {vmatrix} H + \begin {vmatrix} C & A\\F& D \end {vmatrix} I + \begin {vmatrix} B & C\\E& F \end {vmatrix}J = \\ \begin{vmatrix} A & B & C \\ D & E & F\\H&I&J \end{vmatrix} [a×b]⋅c=[ ADBE k+ CFAD j+ BECF i]⋅c= ADBE H+ CFAD I+ BECF J= ADHBEICFJ
混个积代表了以 a × b a \times b a×b为底，斜边为c的菱形体（长方体对吗？）的体积。