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Learn how to express vectors and solve geometry problems using position vectors in a clear and concise manner. Practice identifying parallel lines and midpoint calculations.
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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 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,
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?
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
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
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
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
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!
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