1. Find the magnitude Points A and B have the coordinates (1,4,-4) and (3, 6, 2) respectively. Find the magnitude of the vector

2. Vectors 2 – ParaYell & Vectors of line equations & intersections Know when a vector is parallel with another Understand how to find the vector of a line equation Be able to find intersections of lines in vector form

3. Vector line equations x y o A line can be identified by a linear combination of a position vector and a free vector Any parallel vector (to line) a (any point it passes through) A

4. Vector line equations x y a o (any point it passes through) A line can be identified by a linear combination of a position vector and a free vector A Any parallel vector to line

5. Vector line equations x y o A line can be identified by a linear combination of a position vector and a free vector A parallel vector to line a = xi + yj b = pi + qj E.g. a + b = (xi + yj) + (pi + qj) is a scaler - it can be any number, since we only need a parallel vector

6. Vector Equation of a y = mx + c (1) y = x + 2 1. Position vector to any point on line 1313 [ ] 1313 2. A free vector parallel to the line 2222 [ ] 2222 3. linear combination of a position vector and a free vector xyxy [ ] = + s 1313 [ ] 2222 Equation Scaler (any number)

7. Vector Equation of a y = mx + c (2) y = x + 2 1. Position vector to any point on line 2. A free vector parallel to the line 3. linear combination of a position vector and a free vector Equation Scaler (any number) 4646 [ ] 4646 -3 [ ] -3 [ ] xyxy = + s 4646 [ ] -3 [ ]

8. Vector Equation of a y = mx + c (3) y = 1 / 2 x + 3 1. Position vector to any point on line 2. A free vector parallel to the line 3. linear combination of a position vector and a free vector Equation Scaler (any number) xyxy [ ] = + s 2424 [ ] 2424 4242 4242 4242 2424

9. Sketch this line and find its equation y = 3x - 1 xyxy [ ] = + s 1313 [ ] 1212 1212 1313 1212 = When s=1 xyxy [ ] When s=0 xyxy [ ] = 2525 x=1, y=2 x=2, y=5

10. Finding the equation without a sketch y = 3x - 1 xyxy [ ] = + s 1313 [ ] 1212 1212 When s=0 xyxy [ ] = x=1, y=2 this gives you a coordinate on the line. 1313 [ ] Means go 1 right and 3 up. Gradient = change in y = 3 = 3 change in x 1 Now use y – y 1 = m(x – x 1 ) y – 2 = 3(x – 1) Tells you about the gradient

11. In 2D, line will - Be parallel Intersect Or be the same

12. 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

13. 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

14. Puzzle time

15. Summon the Mathematical Overlord

16. Independent Study Exercise E p 120 (solutions p176) line equations Exercise F p124 (solutions p177) intersections