Geometric meaning of vector cross product

2021-01-12 11:09:59

The geometric meaning of vector cross product

For the two 2 Dimension vector :
a ⃗ = ( x 1 , y 1 ) b ⃗ = ( x 2 , y 2 ) \begin{aligned} \vec{a} &= (x1,y1) \\ \vec{b} &= (x2,y2) \end{aligned} a b =(x1,y1)=(x2,y2)

Definition of cross product :
｜ a ⃗ × b ⃗ ｜ = x 1 y 2 − x 2 y 1 ｜\vec{a} \times \vec{b}｜ = x_1y_2-x_2y_1 a ×b =x1y2x2y1

Calculated area

S = ( x 1 + x 2 ) ( y 1 + y 2 ) S = (x_1+x_2)(y_1+y_2) S=(x1+x2)(y1+y2)

S 1 = x 2 y 1 S_{1} = x_2y_1 S1=x2y1

triangle BFC area :
S 2 = 0.5 ( x 1 y 1 ) S_{2}=0.5(x_1y_1) S2=0.5(x1y1)

triangle OGB area :
S 3 = 0.5 ( x 2 y 2 ) S_{3}=0.5(x_2y_2) S3=0.5(x2y2)

parallelogram OGCA area :
S flat That's ok Four edge shape = S − 2 S 1 − 2 S 2 − 2 S 3 = ( x 1 + x 2 ) ( y 1 + y 2 ) − 2 x 2 y 1 − x 1 y 1 − x 2 y 2 = x 1 y 1 + x 2 y 1 + x 1 y 2 + x 2 y 2 − 2 x 2 y 1 − x 1 y 1 − x 2 y 2 = x 1 y 2 − x 2 y 1 \begin{aligned} S_{ parallelogram } &= S-2S_1-2S_2-2S_3 \\ &= (x_1+x_2)(y_1+y_2) - 2x_2y_1 - x_1y_1 - x_2y_2 \\ &= x_1y_1+x_2y_1 + x_1y_2+x_2y_2 - 2x_2y_1 - x_1y_1 - x_2y_2 \\ &= x_1y_2 - x_2y_1 \end{aligned} S flat That's ok Four edge shape =S2S12S22S3=(x1+x2)(y1+y2)2x2y1x1y1x2y2=x1y1+x2y1+x1y2+x2y22x2y1x1y1x2y2=x1y2x2y1

Conclusion :

The module of vector cross product represents the area of the parallelogram .

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