1.当两个向量均为二维向量时:
a→ = (x1 , y1)\overrightarrow{a}\ =\ \left( x_1\ ,\ y_1 \right) a = (x1 , y1)b→ = (x2 , y2)\overrightarrow{b}\ =\ \left( x_2\ ,\ y_2 \right) b = (x2 , y2)
则,其结果为:
a→×b→= (x1y2 − y1x2)j→\overrightarrow{a}\times \overrightarrow{b}=\ \left( x_1y_2\ -\ y_1x_2 \right) \overrightarrow{j}
a×b= (x1y2 − y1x2)j
该乘积的结果是向量,取其模可以用于由A和B构成的平行四边形的面积,该向量的方向垂直于向量a、b.
2.当两个向量均为三维向量时:
a→ = (x1 , y1, y1)\overrightarrow{a}\ =\ \left( x_1\ ,\ y_1,\ y_1 \right) a = (x1 , y1, y1)b→ = (x2 , y2, z2)\overrightarrow{b}\ =\ \left( x_2\ ,\ y_2 ,\ z_2\right) b = (x2 , y2, z2)
则,可以写为:
∣ijkx1y1z1x2y2z2∣
\left| \begin{matrix}
i& j& k\\
x_1& y_1& z_1\\
x_2& y_2& z_2\\
\end{matrix} \right|
∣∣∣∣∣∣ix1x2jy1y2kz1z2∣∣∣∣∣∣
经行列式计算的结果为:
a→×b→= (y1z2−z1y2 , x2z1−x1z2, x1y2−x2y1)\overrightarrow{a}\times \overrightarrow{b}=\ \left( y_1z_2-z_1y_2\ ,\ x_2z_1-x_1z_2 ,\ x_1y_2-x_2y_1\right)
a×b= (y1z2−z1y2 , x2z1−x1z2, x1y2−x2y1)
3.用行列式计算二维向量叉乘
a→ = (x1 , y1, y1)\overrightarrow{a}\ =\ \left( x_1\ ,\ y_1,\ y_1 \right) a = (x1 , y1, y1)b→ = (x2 , y2, z2)\overrightarrow{b}\ =\ \left( x_2\ ,\ y_2 ,\ z_2\right) b = (x2 , y2, z2)
则,可以写为:
∣ijkx1y10x2y20∣
\left| \begin{matrix}
i& j& k\\
x_1& y_1& 0\\
x_2& y_2& 0\\
\end{matrix} \right|
∣∣∣∣∣∣ix1x2jy1y2k00∣∣∣∣∣∣
经行列式计算的结果为:
a→×b→= (y1∗0−0∗y2 , x2∗0−x1∗0, x1y2−x2y1)\overrightarrow{a}\times \overrightarrow{b}=\ \left( y_1*0-0*y_2\ ,\ x_2*0-x_1*0 ,\ x_1y_2-x_2y_1\right)
a×b= (y1∗0−0∗y2 , x2∗0−x1∗0, x1y2−x2y1)
即为:
a→×b→= ( x1y2−x2y1)k→\overrightarrow{a}\times \overrightarrow{b}=\ \left( \ x_1y_2-x_2y_1\right) \overrightarrow{k}
a×b= ( x1y2−x2y1)k