Sec1Sess2

Section 1 Session 2

Magnitude <math> = |A| = sqrt(a_1^2+a_2^2) </math> which is the euclidean distance from the origin <math> (0,0) </math>

Law of Cosines:
<math>c^2 = a^2 + b^2 - 2*a*b*cos(C)</math>
<math>|\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}|*|\vec{B}|*cos(angle_{ab})</math>
<math>|\vec{C}|^2 = |\vec{C}| . |\vec{C}|</math>
<math>|\vec{C}|^2 = (|\vec{A}| - |\vec{B}|) . (|\vec{A}| - |\vec{B}|)</math>
effect of cos(): <math>(|\vec{A}| . |\vec{B}|) = |\vec{A}|*|\vec{B}|*cos(angle_{ab})</math>

the angle between a vector and itself is 0
cos(0) = 1
<math>|\vec{A}| . |\vec{A}| = |\vec{A}|^2*Cos(angle_{aa})</math>

Combining points into a vector:
NOTE: Order is important, with AB vector from points A and B the calculations are:
<math>\vec{AB} = <b1-a1,b2-a2,c3-a3></math>

DOT PRODUCT
<math>A . B = a_1*b_1 + a_2*b_2</math>
E.g. <math>A = i + 2j, B = 3i + 4j</math>
<math>A . B = 1 * 3 + 2 * 4 = 11</math>
Dot product works the same for 3-dimensional vectors
<math>A . B = a1*b1 + a2*b2 + a3*b3</math>

Test for orthagonality (vectors 90 degrees from each other):
cos(90) = 0, therefore <math>A . B = 0 = A * B * cos(90)</math>

Find length of 3rd triangle side given 2 sides:
Given side A and B, the 3rd side is A-B .
E.x. <math>A = 2i + 3j + 5k</math> and <math>B = i - 2j + 4k,</math> side <math>C = i + 5j + k</math>

Plotted geometric examples of finding a 3rd side are here:
http://tutorial.math.lamar.edu/Classes/CalcII/DotProduct.aspx