Sec1Sess5
complex shapes can be cut up into simpler shapes to find areas
triangle area <math>= 1/2 |A|*|B|*sin(angle) = 1/2 * det(A, B)</math> (see det() description below) where points A, B are relative to origin point C
notice sin() instead of cos()
find sin() by using angle finding methods to find cos() and solving for <math>sin^2()+cos^2()=1</math>
due to complimentary angles <math>sin (angle) = cos(angle')</math>
where <math>angle' = pi/2 - angle. (pi/2 = 90*,</math> the complementary angle<math>)</math>
rotate 90 counter clockwise: vector of <x, y> from origin becomes <-y, x> (yes dimensions are flipped)
Finding determinant:
* KEY TERM * Determinant <math>\begin{bmatrix}a_1 a_2 ; b_1 b_2\end{bmatrix} = (a_1*b_2)-(a_2*b_1)</math>
Det() has same notation as magnitude, | | , but has multiple rows.
Determinant measures the area of a parallelogram with sides A and B
when multiplied by 1/2 the area of a triangle can be found
paral. area will be positive but area of a triangle can be negative
Determinant is + or - the area of a parall. Area is the absolute value of det()
Determinant <math>= |A|*|B|*sin(angle) = |A'|*|B|*cos(angle')</math>
Where A' = <-y, x> of A
A is a vector not a point
= A' . B
= <math><-a_2, a_1>*<b_1, b_2></math>
= <math>(a_1*b_2) - (a_2*b_1)</math>
= det(A, B) where det() is determinant
* IMPORTANT POINT * determinants only work on parallelograms, splitting into triangles is needed to use formulas to find areas on
other types of rectangles.
Calculate 3x3 determinant:
|a b c| = A
|d e f|
|g h i|
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)