1. Vectors (6) Vector Equation of a Line Vector Equation of a Line

2. x y z a Revise: Position Vectors o A In 2D and 3D, all points have position vectors e.g. The position vector of point A a = xi + yj + zk

3. Revise: Parallel Vectors a -10 i + 15 j 2 a -20 i + 30 j 1/5 a1/5 a -2 i + 3 j Vectors with a scaler applied are parallel i.e. with a different magnitude but same direction

4. x y Vector Equation of a line (2D) 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

5. x y a Vector Equation of a line (2D) 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

6. x y Vector Equation of a line (2D) 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 + tb = (xi + yj) + t(pi + qj) t is a scaler - it can be any number, since we only need a parallel vector

7. 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 [ ] = + t 1313 [ ] 2222 Equation Scaler (any number)

8. 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 = + t 4646 [ ] -3 [ ]

9. 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 [ ] = + t 2424 [ ] 2424 4242 4242 4242 2424

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

11. y = 3x - 1 ….. is a Cartesian Equation of a straight line xyxy [ ] = + t 1313 [ ] 1212 ….. is a Vector Equation of a straight line Often written ……. = + t 1313 [ ] 1212 r r is the position vector of any point R on the line Equations of straight lines Any point Direction

12. Convert this Vector Equation into Cartesian form = + t 2525 [ ] 7373 r xyxy = + t 7373 [ ] 2525 Increase in y Increase in x Gradient = the direction vector Gradient (m) = 5 / 2 = 2.5 When t = 0 xyxy [ ] 7373 = x = 7 y = 3 Equations of form y= mx+c y= 2.5x + c 3 = 2.5 x 7 + c c = -14.5 y= 2.5x – 14.5

13. Convert this Vector Equation into Cartesian form (2) = + t 2525 [ ] 7373 r xyxy = + t 7373 [ ] 2525 x = 7 + 2t y = 3 + 5t Convert to Parametric equations Eliminate ‘t’ 5x = 35 + 10t 2y = 6 + 10t subtract 5x – 2y = 29

14. Convert this Cartesian equation into a Vector equation 1414 [ ] Increase in y Increase in x Gradient = Gradient (m) = 4 y = 4x + 3 = + t 1m1m [ ] abab r the direction vector Any point Want something like this ………. When x=0, y = 4 x 0 + 3 = 3 [ ] 0303 = Any point = 4 1 represents the direction = + t 1414 [ ] 0303 r

15. Convert this Cartesian equation into a Vector equation y = 4x + 3 Easier Method Write: y - 3 = 4x = t t = 4x t = y - 3 x = 1 / 4 t y = 3 + t xyxy [ ] = + t 0303 [ ] 1/411/41 = + t 1414 [ ] 0303 r Can replace with a parallel vector

16. Summary = + t 1m1m [ ] abab r the direction vector Any point A line can be identified by a linear combination of a position vector and a free [direction] vector Equations of form y-b=m(x- a) Line goes through (a,b) with gradient m