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Higher Mathematics

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Higher Mathematics

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  1. Higher Mathematics Unit 3.1 Vectors

  2. A B 1.Introduction A vector is a quantity with both magnitude and direction. It can be represented using a direct line segment This vector is named or or u

  3. 3 5 2 2.Vectors in 3 - Dimensions 2 5 3

  4. -2 4 3

  5. 3 2 0

  6. 3 -3 -2

  7. 3. Finding the components of a Vector from Coordinates

  8. P (1, 2) Q (6, 3) 3 2 1 6 6 - 1 5 3 - 2 1

  9. S (-2, 1) T (5, 3) 3 1 -2 5 5 - -2 7 3 - 1 2

  10. A (-2, -1) B (4, 1) 1 - 1 -2 4 4 - -2 6 1- - 1 2

  11. 4.Magnitude

  12. 4 4 -3 -3 42 + (-3)2

  13. 5.Adding Vectors

  14. 7 1 C B 1 2 -6 4 A D Add vectors “ Nose-to-tail”

  15. v u + v u Add vectors “ Nose-to-tail”

  16. A A -u u B B is the negative of is the negative of

  17. v -v u + -v u u - v -2 -4 Add the negative of the vector “ Nose-to-tail”

  18. -v v The Zero Vector Back to the start. Gone nowhere

  19. 7.Multiplication by a Scalar

  20. 2v v 2v has TWICE the MAGNITUDE of v, but v and 2v have the SAME DIRECTION. i.e. They are PARALLEL

  21. 8.Position Vectors

  22. P (4, 2) p The position vector is denoted by If P has coordinates (x , y , z) then the components of the position vector of P are 4 2 The position vector of a point P is the vector from the origin O, to P.

  23. 9.Collinear points

  24. NOT collinear E D then the vectors are parallel and have a point in common - namely B - , this makes them collinear B C Collinear A

  25. 10.Dividing lines in given ratios“Section Formula”

  26. Give up John, they are getting bored!!

  27. 11.Unit Vectors

  28. A unit vector is any vector whose length (magnitude) is one is a unit vector since The vector

  29. There are three special unit vectors:

  30. -2 4 3 +3 +4 -2 All vectors can be represented using a sum of these unit vectors

  31. 12.Scalar Product

  32. The scalar product of the vectors and is defined as: The scalar product (or “dot” product) is a kind of vector “multiplication”. It is quite different from any kind of multiplication we’ve met before. where q is the angle between the vectors, pointing out from the vertex or

  33. Calculating the angle between two vectors We have already seen that Rearranging gives And hence we can find the angle between two vectors

  34. 3. Perpendicular vectors: • Provided and are non zero then if then so ie and are perpendiculiar 2. If either or then 4. Some important results using the scalar product • The scalar product is a number not a vector