Spec_11_Test_3_2018.docx
Tick your teacherMs ChengDr. PearceMs SindelMs RimandoTick your teacherMs ChengDr. PearceMs SindelMs RimandoPERTH MODERN SCHOOL
YR11 MATHEMATICS SPECIALIST – 2018
TEST 3 – Vectors
NAME: _____________________________________ DATE: 14/05/2108 Mark:________
Calculator Assumed Time: 40 minutes Mark: 35 marks
Question 1 (4 marks)
If a= -2i+5j and b=xi-2j are two vectors. Find:
x if a and b are parallel. (2 marks)
x if a and b are perpendicular. (2 marks)
Question 2 (3 marks)
Consider the points A-1,6, B-3,-2 and C (7, 3). Calculate the angle between BA and BC.
Question 3 (4 marks)
Find the scalar projection in the direction of 43o of a vector of magnitude 20 in the direction of 163o.
(2 marks)
Let a=i-j, b=i+3j. Find the vector projection of b in the direction of a. (2 marks)
Question 4 (5 marks)
A triangle is formed by three non-zero vectors a, b and c, so that c=a-b, and θ is the angle between a and b.
Sketch the triangle. (1 mark)
Explain why c∙c=c2. (1 mark)
(c) Use c∙c=a-b∙a-b to deduce the cosine rule. (3 marks)
Question 5 (8 marks)
Consider the quadrilateral in the diagram below.
Use a vector method to
(a) find if ABCD is a parallelogram. (2 marks)
(b) determine the condition(s) required in terms of so that ABCD is a trapezium. (4 marks)
(c) show that points A, B and the origin are collinear. (2 marks)
Question 6 (11 marks)
A small boat that can maintain a steady speed of 5 ms-1 is to cross a river from A to B, where AB=35i-105j m. A current of -i-2j ms-1 flows in the river. The velocity vector that the pilot of the small boat must set to travel from A to B is ai+bj, where a and b are constants.
Explain why ta-1=35 and tb-2=-105, where t is a constant. (3 marks)
Eliminate t from the equations in (a) and hence express b in terms of a, simplifying your expression.
(3 marks)
(c) Explain why a2+b2=25. (1 mark)
(d) Use your equations from (b) and (c) to determine the values of a and b. (3 marks)
(e) Determine the time that the small boat will take to travel from A to B. (1 mark)