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VECTORS

VECTORS. 8 th March 2010. Nigel Copp and John Cooper. Learning Objectives. Vectors position in the National Curriculum Related subjects in mathematics Definition of a vector and examples Vector Notation Vector Operations Ability to complete GCSE vector questions

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VECTORS

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  1. VECTORS 8th March 2010 Nigel Copp and John Cooper

  2. Learning Objectives • Vectors position in the National Curriculum • Related subjects in mathematics • Definition of a vector and examples • Vector Notation • Vector Operations • Ability to complete GCSE vector questions • Vectors within A’Level content

  3. National Curriculum KS4

  4. What is a vector? • A quantity having direction as well as a magnitude (Oxford English Dictionary) • Points in a coordinate system corresponding to points in space (nrich) • Latin origin: Participle stem of vehere "to carry, to convey”. (Wordinfo.com)

  5. Can you think of any examples? • Acceleration: For example Gravity • Force: For example River flow or Wind • Velocity: For example driving a car up the M23 at 70mph • Displacement: For example having got to London from Brighton

  6. A vector may be written in 3 ways...

  7. If we were going from B to A, how would we write it? -a

  8. Addition & subtraction of vectors a~ a~ a~ -b ~ a~ +b ~ b~ b~ a~ -b ~ Vectors must always be added end to end, so that the arrows all point with each other, not against each other.

  9. Addition & subtraction of column vectors -2 4 12 2 5 3 and If = = Then what is 2 - ? a~ b~ a~ b~ -2 4 5 3 - 2 = - 2 = a~ b~

  10. Equal Vectors XY = BZ = Assume that ABC and XYZ are equilateral triangles

  11. Addition & subtraction of vectors B b~ m ~ -a + b + m b + 2m -a + b + 2m -b – m ½ (b – a) + m ½ (a – b) + m ½ (b – a) M AN = NM = AM = OC = AC = MO = BN = O C a~ N A To obtain the unknown vector just get there by any route made up of known vectors.

  12. Consider a frog attempting to swim across a fast running river... 4 m/s 4m 3 m/s Q1: What is the resultant speed and at what angle does the frog cross the river?

  13. Consider a frog attempting to swim across a fast running river... 0 -4 -3 0 -3 -4 + = θ 5m/s 4m/s θ = arctan 3 ÷ 4 = 36.9° 3m/s

  14. Find the overall force from the two Tug Boats... Tug A 10,000 N 40° 20° Tug B 15,000 N

  15. Find the overall force from the two Tug Boats... Tug A 10,000 N 40° 120° 10,000 N 15,000 N R 40° θ° 20° Tug B 15,000 N

  16. The Cosine Rule a2 = b2 + c2 – 2bc COS A The Sine Rule a = b = c . SIN A SIN B SIN C

  17. Some A’level Content • Cartesian Unit Vectors (i and j) 2d and 3d • Vectors in Mechanics • Differentiating and integrating vectors • Scalar Product & Dot Product of 2 vectors • Vector equation of a straight line Some good web site resources for sessions on vectors http://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/vectorshirev1.shtml http://www.mathsphere.co.uk/products/GCSESample2.htm http://www.maa.org/joma/Volume7/Hohenwarter/Transformations.html

  18. Session Plenary. What do we know? • Vectors position in the National Curriculum • Related subjects in mathematics and science • Definition of a vector and examples • Vector Notation and operations • Ability to complete vector questions • Vectors within A’Level content ANY QUESTIONS?

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