1. L.O. Manipulate lines of given vector equations How do Cartesian graphs work? (i.e. y =....) e.g. y = 3x + 2 x can be any number xy 15 28 -3-7 ½3.5

2. L.O. Manipulate lines of given vector equations Consider the equation initial point direction vector (2,-1) t12½-2 direction vector t can be any number

3. L.O. Manipulate lines of given vector equations e.g. find the vector equation of the line through A and B. A (2,-5) B (-3, 8) Direction vector = (or ) Initial point = A (or B) So

4. L.O. Manipulate lines of given vector equations e.g. find the vector equation of the line through: 1. 2. A (5, -3) and B (-4, 2) 3.

5. L.O. Manipulate lines of given vector equations e.g. find the vector equation of the line through (-1, 0) and parallel to y = 4x - 2

6. L.O. Manipulate lines of given vector equations Does (5, -36) lie on ? A (2,-5) B (-3, 8) i.e. is there some value of t which will generate the point (5, -36)? If it does: t cant be both things at once – so the point is not on the line.

7. L.O. Manipulate lines of given vector equations Does (3, 2) lie on Does (1, -3) lie on

8. L.O. Manipulate lines of given vector equations Parametric form – each dimension (x, y, z) given in terms of a parameter λ e.g. r = i + 2j – 4k + λ(6i – 7j + 3k)

9. L.O. Manipulate lines of given vector equations Find the parametric equations of the line passing through A (-2, 1, 3) and B (1, -1, 4)

10. L.O. Manipulate lines of given vector equations Cartesian coordinates from parametric x = 3μ + 1 y = 2μ – 1 z = 4 + μ

11. L.O. Manipulate lines of given vector equations Cartesian coordinates from parametric x = 3μ + 1 y = 2μ – 1 z = 4 + μ

12. L.O. Manipulate lines of given vector equations Cartesian coordinates from parametric x = 3μ + 1 y = 2μ – 1 z = 4 + μ

13. L.O. Manipulate lines of given vector equations Cartesian coordinates from parametric x = 3μ + 1 y = 2μ – 1 z = 4 + μ

14. L.O. Manipulate lines of given vector equations Cartesian coordinates from parametric x = 3μ + 1 y = 2μ – 1 z = 4 + μ

15. L.O. Manipulate lines of given vector equations Cartesian coordinates from parametric x = 3μ + 1 y = 2μ – 1 z = 4 + μ

16. L.O. Manipulate lines of given vector equations A line is parallel to the vector 2i – j + 2k and passes through (2, -3, 5). Find: the vector equationthe parametric equation the Cartesian equation

17. L.O. Manipulate lines of given vector equations Are these lines parallel? Consider the direction vectors. Therefore, are parallel.

18. L.O. Manipulate lines of given vector equations Are these lines parallel? so not parallel. Consider parametric form: r1:r2:r1:r2: x = λx = 2 y = - λy = 1 + 3μ z = 1 - 3λz = 5 μ Do they intersect? If so x = x y = yz = z λ = 2-λ =1 + 3μ 1 - 3λ = 5μ -2 = 1 + 3μ -1 = μ 1 – 6 = -5 Equation systems work intersect, at: x = λ = 2 y = - λ= -2 (2, -2, -5) z = 1 - 3λ= -5 Solve the simultaneous equation set. If you can find values for λ and μ that work, then they intersect. If, for example, this line was 1 – 6 = 10, the equation set is inconsistent and cannot be solved, so the lines do not intersect. (are skew) Use either value in the respective equation to find the point of intersection.

19. L.O. Manipulate lines of given vector equations To find the angle between two vectors, use the formulae from the book for the product a.b (Method 1 – Easier) e.g.

20. L.O. Manipulate lines of given vector equations To find the angle between two vectors, use the formulae from the book for the product a.b (Method 2 – Harder) a.b = a 1 b 1 + a 2 b 2 + a 3 b 3 = |a|. |b|. cosθ e.g. a.b = 2x5 + -1x2 =.. 10 - 2 = 5. 29. cosθ