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Vector Diagrams. F. L. R. X. D’. K’. E. K. Q. W. C’. J’. D. J. P. V. B’. I’. C. I. O. U. A’. H’. B. H. N. T. Z. G’. a. A. G. M. S. Y. F’. b. Write the following in terms of a and b

jeffreyfletcher
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  1. Vector Diagrams F L R X D’ K’ E K Q W C’ J’ D J P V B’ I’ C I O U A’ H’ B H N T Z G’ a A G M S Y F’ b Write the following in terms of a and b AY, MR, CP, VF, US, PH, C’U, LG, AY = 4b, MR = 5a, CP = a + 2b, VF = 2a – 3b, US = -2a, PH = -2a - b, C’U = -2a - b, LG = -5a,

  2. B C ABCD is a parallelogram, Let AB = a And AD = b M is the mid-point of AD a X M D A b What are the following in terms of a and b: AC, BD, DB, AM, BX, AX, MX, AC = AB + BC = a + b BX = ½ BD = - ½ a + ½ b BD = BA + AD = -a + b AX = ½ AC = ½ a + ½ b MX = MA + AX = -½ b + (½a + ½b) = ½ a DB = -BD = a - b AM = ½ AD = ½ b Which answers are parallel?

  3. P D C ABCD is a parallelogram, MNPQ are the midpoints of the sides, as shown Let MQ = x And AM = y Q N B A M Express in terms of x and y: AB, AQ, NB, BC, AC, BD AB = 2y AQ = AM + MQ = y + x NB = -AQ = - y - x BC = 2AQ = 2y + 2x AC = AB + BC = 2y + (2y + 2x) = 4y + 2x BD = BC + CD = (2y + 2x)- 2y = 2x

  4. X Y Z O OX = 3a + 3b Not drawn to scale OY = 5a + 2b OZ = 6a Express in terms of a and b: XY, YZ and XZ XY = -OX + OY = 2a - b YZ = -OY + OZ = a - 2b XZ = -OX + OZ = 3a – 3b What facts does this tell you about triangle OXZ ? Isosceles and length of OX = length of XZ

  5. The midpoint of vector a + b A If A and B have position vectors a and b, then the position vector of the midpoint of line AB is ½ (a + b) M O B This result is often useful. Look for parallelograms in a vector problem

  6. A vector joining the origin to a point is called a position vector. The position vectors of X, Y and Z are OX, OY and OZ X Y Let these vectors be x, y and z then: O Z XY = -x + y YX = -y + x ZY = -z + x XZ = -x + z

  7. A B C O The position vectors of A, B and C are a, b and 4b – 3a respectively Find AB in terms of a and b AB = AO + OB = -a + b Show clearly that A, B and C lie in a straight line AC = AO + OC = -a + (4b – 3a) = 4b – 4a = 4(b – a) = 4AB AC and AB are parallel and both go through A So they must be on the same straight line!

  8. B A a b F C O E D Exam question ABCDEF is a regular hexagon, Let OA = a And OB = b Write down, in terms of a and b, the vectors i) AB and ii) FC iii) Write down one geometrical fact about AB and FC which could be deduced from your answer i) AB = b – a iii) AB and FC are parallel since FC = 2AB ii) FC = 2b - 2a

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