1. Vectors (8) 2D Vector Equations 2D Vector Equations point of intersectpoint of intersect 3D Vector Equations 3D Vector Equations do they intersectdo they intersect

2. In 2D : lines either ……. Are parallel intersect Or are the same

3. Intersect of 2D lines in vector form - 1 xyxy [ ] = + s 1313 [ ] 2222 xyxy = + t 6 -2 [ ] 4 [ ] and For example If the lines intersect, there must be values of s and t that give the position vector of the point of intersection. x part: 1 + 2s = 6 - t y part: 3 + 2s = -2 + 4t Subtract x from y : 2 = -8 + 5t 5t = 10 t = 2 Substitute: 1 + 2s = 6 - 2 2s = 3 s = 1.5 xyxy [ ] = + 1.5 1313 [ ] 2222 xyxy = + 1313 [ ] 3333 xyxy = 4646 position vector of the point of intersection

4. Intersect of 2D lines in vector form - 2 r = (i + 2j) + (4i - 2j) s = (2i - 6j) +  (-2i + j) Point of intersection? i coefficients : 1 + 4 = 2 -2  j coefficients : 2 - 2 = -6 +  x2 : 4 - 4 = -12 + 2  add 5 = -10 … doesn’t work Direction vectors: (4i - 2j) and (-2i + j) are parallel ….. lines are parallel

5. Example The lines r and s have the equations... and Show they intersect and find the point of intersection If the lines intersect, there must be values of and  that give the position vector of the point of intersection. x : 2 + 4 = 4 +2  y : 3 - = 7 - 2  z : 5 + 3 = 2 +3  x+y : 5 + 3 = 11 3 = 6 = 2 Substitutex : 2 + 4 x 2 = 4 +2   = 3 Check the values in the 3rd equation z : 5 + 3 x 2 = 2 +3 x 3 11 = 11 Satisfied! Hence a point exists common to both lines

6. Example (continued) The lines r and s have the equations... and Show they intersect and find the point of intersection If the lines intersect, there must be values of and  that give the position vector of the point of intersection. = 2  = 3 Values satisfy all equations Hence a point exists common to both lines Substitute The 2 lines intersect at (10, 1, 11)