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Warm Up

Warm Up. Find the distance between (-3, -5) and (-6, 2). 14a Vectors. Photo: http://villains.wikia.com/wiki/Vector_%28Despicable_Me%29. Vectors. Vectors are quantities which are fully described by both a magnitude (size) and a direction .

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Warm Up

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  1. Warm Up Find the distance between (-3, -5) and (-6, 2)

  2. 14a Vectors Photo: http://villains.wikia.com/wiki/Vector_%28Despicable_Me%29

  3. Vectors • Vectors are quantities which are fully described by both a magnitude (size) and a direction. • Examples include force, momentum, velocity and displacement.

  4. Writing about vectors… B • This vector could be represented by or a or a A The magnitude (length) is represented by: or AB or |a| or

  5. Vectors starting at the origin, O A a • This vector could be represented by • or a • or a O The magnitude (length) could be represented by: or OA or |a|or

  6. a b c Vector Equality • Two vectors are equal if they have the same magnitude and direction. • They will be parallel and equal in length.

  7. A B Negative Vectors • …have the same length but opposite directions.

  8. 1) see page 375 PQRS is a parallelogram, and PQ = a and QR = b. b Q R a S P Find vector expressions for:

  9. see page 375 1) PQRS is a parallelogram, and PQ = a and QR = b. b Q R a S P Find vector expressions for:

  10. 14b Operations with Vectors

  11. Bearings.. • Bearings are always measured from the northand in a clockwise direction • A direction of 115º …. • A direction of 015º ….

  12. Nov 2008, paper 2, #6

  13. a a a b b b Vector addition • to add a and b • draw a • at the arrowhead (tip) of a draw b • join the beginning (tail) of a to the tip of b a + b The vectors are always added from tip to tail.

  14. A runner runs in an easterly direction for 4 km and then in a southerly direction for 2 km. How far is she from her starting point and in what direction? This vector is called the displacement vector. 4km θ 2km x km

  15. Suppose we have three towns P, Q and R. A trip from P to Q followed by a trip from Q to R is equivalent to a trip from P to R. In vector form: Q P Q P R Q R P R

  16. P. 376 Example 4

  17. B A E C D 2) Find a single vector which is equal to:

  18. B A E C D 2) Find a single vector which is equal to:

  19. Zero Vector • The zero vector is written as 0 and for any vector a, • a + (-a) = (-a) + a = 0

  20. Vector Subtraction • To subtract one vector from another, we simply add its negative: • a – b = a + (-b)

  21. -b a b b -b Vector Subtraction • To subtract one vector from another, we simply add its negative: • a – b = a + (-b) a a - b

  22. 3) a) r – s b) s – t – r P. 378 For r, s and t as shown find geometrically: a) r – s b) s – t – r r s t r -s r – s -r -t s – t – r s

  23. Based on the diagram… y Q P z x R x = y + z z = -y + x y = x + -z

  24. 4) For points A, B, C and D, simplify the following vector expressions:

  25. For points A, B, C and D, simplify the following vector expressions: 4) =AB + -CB =AB - -BC =AB + BC = =AC + CB – DB =AB – DB =AB + BD =

  26. see page 379 5) Construct vector equations for:

  27. 6) Find in terms of r, s and t:

  28. P.380

  29. Scalars • … are quantities which are fully described by a magnitude alone (no direction). • Examples are time, speed, distance

  30. Scalar Multiplication • If a is a vector, what would 2a and -3a mean? • 2a = a + a • 3a = a + a + a • -3a = 3(-a) = -a + -a + -a -a a a -a a 2a -a -3a

  31. Given vectors r and s. Geometrically find: a) 2r + s b) r – 3s b) r – 3s a) 2r + s 7) r s

  32. Homework • p.375 #1, 2 • p.376 #1ac, 2 • p.378 #1ac, 3, 4 • p.381 #2 • p. 382 #1ae • HW Quiz next class

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