1 Chalkboard
Figure 1: Area within 2 vectors
Figure 2: Rotated vector \(\vec{A}\)
Figure 3: Area as determinant of 2 vectors
Figure 4: Area of parallelogram and triangle
2 What is the area between 2 vectors?
2.1 Front
What is the area between 2 vectors?
2.2 Back
- Length and determinant
- \(\vb{A’} = \vb{A}\) rotated \(\SI{90}{\degree}\)
- \(\theta’ = \pi/2 - \theta\)
- \(\cos \theta’ = \sin \theta\)
- Area of parallelogram
- \(\abs{\vb{A}} \abs{\vb{B}} \sin \theta = \abs{\vb{A’}} \abs{\vb{B}} \cos \theta’ = \vb{A’} \cdot \vb{B} = \ev{-a_2, a_1} \cdot \ev{b_1, b_{2}} = a_1b_2 - a_2b_1 = \det(\vb{A}, \vb{B})\)
- absolute value of determinant of \(\vb{A}\) and \(\vb{B}\)
- Area of triangle
- \({\displaystyle \abs{\frac{1}{2} \det(\vb{A}, \vb{B})}}\)
- Cross product
- Area of parallelogram
- \(\abs{\vb{A} \cross \vb{B}}\)
- Area of triangle
- \({\displaystyle \frac{1}{2} \abs{\vb{A}} \cross \abs{\vb{B}}}\)
- Area of parallelogram