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Mathematics Extension 1

ME-V1 Introduction to vectors

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Table of contentsMathematics Extension 1...........................................................................1ME-V1 Introduction to vectors....................................................................1

Table of contents.......................................................................................................................................... 2Outcomes..................................................................................................................................................... 4Content......................................................................................................................................................... 4Supplementary resources............................................................................................................................ 7

Department of Education resources.........................................................................................................7NESA resources...................................................................................................................................... 7Jonathan Kim Sing videos....................................................................................................................... 7

Examination-style questions......................................................................................................................... 8Sample question 1................................................................................................................................... 8Sample question 2................................................................................................................................... 8Sample question 3................................................................................................................................... 9Sample question 4................................................................................................................................... 9Sample question 5................................................................................................................................. 10Sample question 6................................................................................................................................. 10Sample question 7................................................................................................................................. 11Sample question 8................................................................................................................................. 12Sample question 9................................................................................................................................. 13Sample question 10............................................................................................................................... 13Sample question 11............................................................................................................................... 14Sample question 12............................................................................................................................... 15Sample question 13............................................................................................................................... 16Sample question 14............................................................................................................................... 17Sample question 15............................................................................................................................... 18Sample question 16............................................................................................................................... 19Sample question 17............................................................................................................................... 20

Solutions.................................................................................................................................................... 21Sample question 1................................................................................................................................. 21Sample question 2................................................................................................................................. 21Sample question 3................................................................................................................................. 22Sample question 4................................................................................................................................. 22Sample question 5................................................................................................................................. 23Sample question 6................................................................................................................................. 23Sample question 7................................................................................................................................. 24Sample question 8................................................................................................................................. 24Sample question 9................................................................................................................................. 24Sample question 10............................................................................................................................... 25Sample question 11............................................................................................................................... 25Sample question 12............................................................................................................................... 26Sample question 13............................................................................................................................... 27Sample question 14............................................................................................................................... 28Sample question 15............................................................................................................................... 29Sample question 16............................................................................................................................... 30Sample question 17............................................................................................................................... 30

2 ME-V1 Introduction to vectors

Disclaimer

This document is to be used to supplement the support teachers are offering students undertaking HSC Mathematics courses. Questions can be printed off for students individually, with or without solutions, or as an entire booklet. Questions have been sourced from various states across Australia and the source of each question has been referenced. Permission to use these resources was provided in June 2020. Solutions for each of the questions can be found at the end of the document.

Outcomes

All outcomes referred to in this booklet are from Mathematics Extension 1 Syllabus © 2017 NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales.

© NSW Department of Education, Sep-2320

OutcomesA student:› applies concepts and techniques involving vectors and projectiles to solve problems

ME12-2› chooses and uses appropriate technology to solve problems in a range of contexts

ME12-6› evaluates and justifies conclusions, communicating a position clearly in appropriate

mathematical forms ME12-7

ContentV1.1: Introduction to vectors

Students:

● define a vector as a quantity having both magnitude and direction, and examine examples of vectors, including displacement and velocity (ACMSM010)o explain the distinction between a position vector and a displacement (relative)

vector define and use a variety of notations and representations for vectors in two dimensions

(ACMSM014)

o use standard notations for vectors, for example: a~ , AB and a

o represent vectors graphically in two dimensions as directed line segmentso define unit vectors as vectors of magnitude 1, and the standard two-dimensional

perpendicular unit vectors i~ and

j~

o express and use vectors in two dimensions in a variety of forms, including component form, ordered pairs and column vector notation

● perform addition and subtraction of vectors and multiplication of a vector by a scalar algebraically and geometrically, and interpret these operations in geometric terms AAM o graphically represent a scalar multiple of a vector (ACMSM012)o use the triangle law and the parallelogram law to find the sum and difference of two

vectorso define and use addition and subtraction of vectors in component form (ACMSM017) o define and use multiplication by a scalar of a vector in component form

(ACMSM018)

4 ME-V1 Introduction to vectors

V1.2: Further operations with vectors

Students:

● define, calculate and use the magnitude of a vector in two dimensions and use the

notation ~u

for the magnitude of a vector u~=x i

~+ y j

~

o prove that the magnitude of a vector, u~=x i

~+ y j

~ , can be found using:

|u~|=|x i

~+ y j

~|=√x2+ y2

o identify the magnitude of a displacement vector AB as being the distance between the points A and B

o convert a non-zero vector u~ into a unit vector

u~ by dividing by its length:

u~=

u~

|u~|

● define and use the direction of a vector in two dimensions

● define, calculate and use the scalar (dot) product of two vectors u~=x1 i

~+ y1 j

~ and v~=x2 i

~+ y2 j

~ AAM

o apply the scalar product, u~⋅v

~ , to vectors expressed in component form, whereu~⋅v

~=x1 x2+ y1 y2

o use the expression for the scalar (dot) product, u~⋅v

~=|u

~||v

~|cosθ

where θ is the angle

between vectors u~ and

v~ to solve problems

o demonstrate the equivalence, u~⋅v

~=|u

~||v

~|cosθ=x1 x2+ y1 y2

and use this relationship to solve problems

o establish and use the formula v~⋅v

~=|v

~|2

o calculate the angle between two vectors using the scalar (dot) product of two vectors in two dimensions

● examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021)

● define and use the projection of one vector onto another (ACMSM022)● solve problems involving displacement, force and velocity involving vector concepts in

two dimensions (ACMSM023) AAM

© NSW Department of Education, Sep-2320

● prove geometric results and construct proofs involving vectors in two dimensions including to proving that: AAM o the diagonals of a parallelogram meet at right angles if and only if it is a rhombus

(ACMSM039)o the midpoints of the sides of a quadrilateral join to form a parallelogram

(ACMSM040)o the sum of the squares of the lengths of the diagonals of a parallelogram is equal to

the sum of the squares of the lengths of the sides (ACMSM041)

6 ME-V1 Introduction to vectors

Supplementary resourcesDepartment of Education resources

Units of work ME-V1 Introduction to vectors

ME V1 vectors example solutions

HSC Hub videos Determining the projection of one vector on to another Q12b NESA sample

examination

Finding the angle between vectors Q1 NESA sample examination

Using vector arithmetic Q2 NESA sample examination

NESA resources Mathematics Extension 1 – Sample examination materials (2020)

Jonathan Kim Sing videos Vectors (Y12 Extension 1)

© NSW Department of Education, Sep-2320

Examination-style questionsSample question 1

Consider the vectors given by a=mi+ j and b=i+m j, where m∈ R.

If the acute angle between is 30°, then m equals

a) √2±1

b) 2±√3

c) √3 , 1√3

d) √34−√3

e) √3913

Source: Question 11 ©VCAA 2018 VCE Specialist mathematics written examination 2

Sample question 2

A body has displacement of 3 i+ j metres at a particular time. The body moves with

constant velocity and two seconds later its displacement is −i+5 jmetres.

The velocity, in ms–1, of the body is

a) 2 i+6 j

b) −2 i+2 j

c) −4 i+4 j

d) 4 i−4 j

e) i+3 j

Source: Question 15 ©VCAA 2017 VCE Specialist mathematics written examination 2

8 ME-V1 Introduction to vectors

Sample question 3A force of magnitude 4 newtons acts in the north-easterly direction and another force of 3 newtons acts in the easterly direction.

The magnitude, in newtons, of the resultant of these two forces is

a) 7

b) 25+12√2

c) √25+12√2

d) √25−12√2

e) 25−12√2

Source: Question 15 ©VCAA 2009 VCE Specialist mathematics written examination 2

Sample question 4Consider the three forces

F1=−√3 j, F2=i+√3 j and F3=−12

i+ √32

j

The magnitude of the sum of these three forces is equal to

a) the magnitude of (F3−F1)

b) the magnitude of (F ¿¿2−F1)¿

c) the magnitude of F1

d) the magnitude of F2

e) the magnitude of F3

Source: Question 11 ©VCAA 2011 VCE Specialist mathematics written examination 2

© NSW Department of Education, Sep-2320

Sample question 5A particle is acted on by two forces, one of 6 newtons acting due south, the other of 4 newtons acting in the direction N 60°W .

The magnitude of the resultant force, in newtons, acting on the particle is

a) 10

b) 2√7

c) 2√19

d) √52−24 √3

e) √52+24√3

Source: Question 14 ©VCAA 2012 VCE Specialist mathematics written examination 2

Sample question 6Forces of magnitude 5 N, 7 N and Q N act on a particle that is in equilibrium, as shown in the diagram below.

The magnitude of Q, in newtons, can be found by evaluating

a) √52+72−2×5×7 cos (70° )

b) 52+72−2×5×7cos (110° )

c) √52+72−2×5×7 cos (110° )

d) 52+72−2×5×7cos (70 ° )

e) √52+72−2×5×7 cos (20 ° )

Source: Question 16 ©VCAA 2013 VCE Specialist mathematics written examination 2

10 ME-V1 Introduction to vectors

© NSW Department of Education, Sep-2320

Sample question 7

A body on a horizontal smooth plane is acted upon by four forces, F1, F2, F3 and F4, as

shown.

The force F1 acts in a northerly direction and the force F4 acts in a westerly direction.

Given that |F1|=1, |F2|=2, |F3|=4 and |F4|=5, the motion of the body is such that it

a) is in equilibrium.

b) moves to the west.

c) moves to the north.

d) moves in the direction 30° south of west.

e) moves to the east.

Source: Question 18 ©VCAA 2014 VCE Specialist mathematics written examination 2

12 ME-V1 Introduction to vectors

Sample question 8Question 3 (8 marks)

Figure 2 shows the quadrilateral ABCD, where AB=a, BC=b, and CD=c.

The points E, F, G, and H are the midpoints of the sides AB, BC, CD, and DA respectively.

Figure 2

a) Find the following vectors in terms of a, b, and c.

(i) AD (1 mark)

(ii) EF (1 mark)

(iii) HG (2 marks)

b) Explain why EFGH is a parallelogram.

(2 marks)

Source: Sample question 3 ©South Australian Certificate of Education VCE Specialist Mathematics 2019

© NSW Department of Education, Sep-2320

Sample question 9a) Figure 1 shows rhombus OPQR with OP= p and ¿=r

Figure 1(i) Find OQ in terms of p and r.

(1 mark)(ii) Find PR in terms of p and r.

(1 mark)

(iii) Show that OQ . PR=|r|2−|p|2, hence prove that the diagonals of the rhombus OPQR are perpendicular, giving reasons.

(4 marks)

Source: Sample question 3 ©South Australian Certificate of Education VCE Specialist Mathematics 2017

Sample question 10

A particle moves in the cartesian plane with position vector r=x i+ y j where x and y are

functions of t . If its velocity vector is v=− y i+x j, find the acceleration vector of the particle

in terms of the position vector r.

(3 marks)

Source: Question 9 ©South Australian Certificate of Education VCE Specialist Mathematics 2007

14 ME-V1 Introduction to vectors

Sample question 11The point O is on a sloping plane that forms an angle of 45 ° to the horizontal. A particle is projected from the point O. The particle hits a point A on the sloping plane as shown in the diagram.

The equation of the line OA is y=−x. The equations of motion of the particle are

x=18 t

y=18√3 t−5 t2

where t is the time in seconds after projection. Do NOT prove these equations.

(i) Find the distance OA between the point of projection and the point where the particle hits the sloping plane.

(2 marks)(ii) What is the size of the acute angle that the path of the particle makes with the

sloping plane as the particle hits the point A?

(3 marks)

Source: Question 13d ©NSW Education Standards Authority HSC Mathematics Extension 1 2019

© NSW Department of Education, Sep-2320

Sample question 12An object is projected from the origin with an initial velocity of V at an angle q to the horizontal. The equations of motion of the object are

x (t )=Vt cosθ

y ( t )=Vt sin θ− g t2

2.

(Do not prove these)(i) Show that when the object is projected at an angle θ, the horizontal range is

V 2

gsin 2θ

(2 marks)

(ii) Show that when the object is projected at an angle π2−θ, the horizontal range is

also V2

gsin 2θ

(1 mark)

The object is projected with initial velocity V to reach a horizontal distance d, which is less than the maximum possible horizontal range. There are two angles at which the object can be projected in order to travel that horizontal distance before landing.

Let these angles be α and β, where β=π2−α.

Let hα be the maximum height reached by the object when projected at the angle α to the horizontal.

Let hβ be the maximum height reached by the object when projected at the angle β to the horizontal.

16 ME-V1 Introduction to vectors

(iii) Show that the average of the two heights, hα+hβ

2, depends only on V and g.

(3 marks)

Source: Question 13c © NSW Education Standards Authority HSC Mathematics Extension 1 2018

Sample question 13A golfer hits a golf ball with initial speed V ms-1 at an angle θ to the horizontal. The golf ball is hit from one side of a lake and must have a horizontal range of 100 m or more to avoid landing in the lake.

Neglecting the effects of air resistance, the equations describing the motion of the ball are

x=Vt cosθ

y=Vt sinθ−12g t2

where t is the time in seconds after the ball is hit and g is the acceleration due to gravity in ms−2. Do NOT prove these equations.

(i) Show that the horizontal range of the golf ball is V2sin 2θg

metres

(2 marks)(ii) Show that if V 2<100g then the horizontal range of the ball is less than 100 m.

(1 mark)(iii) It is now given that V 2=200 g and that the horizontal range of the ball is 100 m or

more.

Show that π12

≤θ≤ 5 π12

(2 marks)

© NSW Department of Education, Sep-2320

(iv) Find the greatest height the ball can achieve.

(2 marks)

Source: Question 13c ©NSW Education Standards Authority HSC Mathematics Extension 1 2017

Sample question 14The trajectory of a projectile fired with speed u ms–1 at an angle θ to the horizontal is represented by the parametric equations

x=ut cos θ and y=ut sin θ−5 t 2

where t is the time in seconds.

(i) Prove that the greatest height reached by the projectile is u2sin 2θ

20.

(2 marks)

A ball is thrown from a point 20 m above the horizontal ground. It is thrown with speed 30 ms–1 at an angle of 30 ° to the horizontal. At its highest point the ball hits a wall, as shown in the diagram.

(ii) Show that the ball hits the wall at a height of 1254 m above the ground.

(2 marks)

The ball then rebounds horizontally from the wall with speed 10 ms–1. You may assume that the acceleration due to gravity is 10 ms–2.

(iii) How long does it take the ball to reach the ground after it rebounds from the wall?

18 ME-V1 Introduction to vectors

(2 marks)(iv)How far from the wall is the ball when it hits the ground?

(1 mark)

Source: Question 13b ©NSW Education Standards Authority HSC Mathematics Extension 1 2016

Sample question 15A projectile is fired from the origin O with initial velocity V ms−1 at an angle θ to the horizontal. The equations of motion are given by

x=Vt cosθ, y=Vt sinθ−12g t2. (Do NOT prove this.)

(i) Show that the horizontal range of the projectile is V2sin 2θg

(2 marks)

A particular projectile is fired so that θ=π3

(ii) Find the angle that this projectile makes with the horizontal when t=2V√3 g

(2 marks)

(iii) State whether this projectile is travelling upwards or downwards when t=2V√3 g

. Justify

your answer.

(1 mark)

Source: Question 14a ©NSW Education Standards Authority HSC Mathematics Extension 1 2015

© NSW Department of Education, Sep-2320

20 ME-V1 Introduction to vectors

Sample question 16The diagram below shows a quadrilateral ABCD. Points E, F, G and H are midpoints of the intervals AB, BC, CD and DA respectively.

(i) Use vector methods to show EF ∥ AC.

(1 mark)(ii) Hence prove that EFGH is a parallelogram.

(2 marks)

Source: NSW Educations Standards Authority, Mathematics Extension 1 Stage 6 syllabus, Page 51

© NSW Department of Education, Sep-2320

Sample question 17This question has been designed by the Mathematics Curriculum Support team. It illustrates an application of the scalar product.

The diagram below shows a triangle ABC, where AB=x1 i+ y1 j and AC=x2i+ y2 j.

(i) Show that vector w= y2 i−x1 j is perpendicular to AC .

(1 mark)

(ii) Show that the scalar projection of AB on to w is given by x1 y2−x2 y1

√x22+ y2

2

(2 marks)(iii) Hence find the area of the triangle ABCD in terms of x1, y1, x2 and y2.

(1 mark)(iv)Find the area of the triangle PQR, where P(3 ,5), Q(5 ,7) and R(8 ,2)

(2 marks)

22 ME-V1 Introduction to vectors

SolutionsSample question 1

|a|=√m2+1 and |b|=√m2+1

a .b=|a||b|cosθ

m+m=(√m2+1) (√m2+1)cos30

0=√3m2−4m+√3

m=−4 ±√(−4 )2−4 (√3 ) (√3 )

2√3

Answer = c) √3 , 1√3

Sample question 2

A body has displacement of 3 i+ j metres at a particular time. The body moves with

constant velocity and two seconds later its displacement is −i+5 jmetres.

The velocity, in ms–1, of the body is

Change∈displacement=−4 i+4 j

Change∈time=2 seconds

Therefore constant velocity=Change∈displacement

Change∈time

Answer = b) −2 i+2 j

© NSW Department of Education, Sep-2320

Sample question 3

|R|2=32+42−2×3×4× cos135

|R|=√25+ 12√2

Answer = c) √25+12√2

Sample question 4

F1+F2+F3=(−√3 j)+( i+√3 j)+(−12

i+ √32

j)=12i+ √3

2j

Answer = e) the magnitude of F3

24 ME-V1 Introduction to vectors

Sample question 5

|R|2=62+42−2×6×4× cos60

|R|=√52−24

Answer = b) 2√7

Sample question 6

© NSW Department of Education, Sep-2320

Answer = a) √52+72−2×5×7 cos (70° )

Sample question 7

Resolving horizontally ⟶ positive

Resultant R x=2 sin 60+4 sin 60−5=3√3−5=0.196 N

Resolving vertically ↑ positive

Resultant R y=1+2cos 60−4cos60=0

Answer = e) moves to the east.

Sample question 8a) i. a+b+c

ii.12(a+b)

iii.12AD−1

2c=1

2(a+b+c )−1

2c=1

2(a+b)

b) EF=12(a+b) and HG=1

2(a+b), therefore EF ∥HG

Similarly FG=12(b+c) and EH=1

2(b+c), therefore FG∥EH and EFGH is a parallelogram

Sample question 9a) i. r+ p

ii. r−pb) OQ . PR=(r+ p ) . (r−p )=r .r−p . p=|r|2−|p|2

26 ME-V1 Introduction to vectors

Rhombus, therefore |r|=|p| and OQ . PR=0.

OQ and PR are the diagonals of the rhombus and are perpendicular as OQ . PR=0

Sample question 10

dxdt

=− y⇒ d2 xd t2 =

−dydt , dydt

=x⇒ d2 yd t 2 =dx

dt

∴ d2 xd t 2 =−x and d

2 yd t2 =− y

Acceleration a=−x i− y j=−r

Sample question 11i. 18√3t−5 t 2=−18 t

∴t=0 or t=18 (√3+1 )5

x=18. 18 (√3+1)5

=250.37 m

ii. x.

=18 and y.

=18√3−10¿

tan α= 1836+18√3

, α=15 °

Therefore the angle with the sloping plane is equal to 45 °−15°=30°

Source: Question 13d ©NSW Education Standards Authority HSC Mathematics Extension 1 marking guidelines 2019

© NSW Department of Education, Sep-2320

28 ME-V1 Introduction to vectors

Sample question 12

i. Vt sin θ−g t 2

2=0

t=2V sin θg , as t>0

x=V ( 2V sinθg )cosθ

x=V 2

gsin 2θ, as sin 2θ=2sin θ cosθ

Therefore the range, at angle θ, is V2

gsin 2θ

ii. Range, R=V 2

gsin 2( π2 −θ)

R=V 2

gsin (π−2θ)⇒R=V 2

gsin 2θ

iii. y.

(t )=0, V sin θ−¿=0, t=V sin θ

g

y=V 2 sin2θg

−V 2 sin2θ2 g

=V 2 sin2θ2 g

Therefore hα=V 2 sin2α

2g and hβ=

V 2 sin2 β2g

but sin β=sin ( π2 −α)=cos α

Therefore hα+hβ

2=

V 2 sin2α2g

+V2sin2 β2g

2=V 2sin2α+V 2cos2α

4 g= V 2

4 g which depend on V and g only.

Source: Question 13c ©NSW Education Standards Authority HSC Mathematics Extension 1 marking guidelines 2018

© NSW Department of Education, Sep-2320

Sample question 13

i. Vt sinθ−12g t 2=0

t=2V sin θg , as t>0

x=V ( 2V sinθg )cosθ=V 2

gsin 2θ

ii. If V 2<100g then Range¿100g sin 2θ

g

and Range¿100 as sin 2θ<1

iii. If V 2=200 g then range ¿200g sin 2θ

g≥100 and sin 2θ≥ 1

2 where 0≤θ≤π2 , therefore 0≤2θ≤π

π6≤2θ≤ 5 π

6 and π12

≤θ≤ 5 π12

Source: Question 13c ©NSW Education Standards Authority HSC Mathematics Extension 1 marking guidelines 2017

30 ME-V1 Introduction to vectors

Sample question 14

i. y.

=u sin θ−10 t, max height when y.

=0

Therefore 0=u sin θ−10 t , t=u sinθ

10

Max height, ymax, when t=u sinθ

10

Therefore ymax=u( u sinθ10 )sinθ−5( u sinθ

10 )2

¿ u2sin 2θ10

−u2 sin2θ20

¿ u2sin 2θ20

, maximum height is u2sin 2θ20

ii. Max height ¿20+ 302 sin23020

=1254

iii. y..

=−10

y.

=−10t , as y.

(0 )=0

y=−5t2, as y (0 )=0

Ball hits the ground at y=−125

4

−1254

=−5 t2, t=52=2.5 seconds as t>0

Therefore the ball hits the ground 2.5 seconds after hitting the walliv. x

.

=10, taking left of the wall as positive.

x=10 t , as x (0 )=0

x=10×2.5=25 metres.

The ball lands 25 metres from the wall.

Source: Question 13b ©NSW Education Standards Authority HSC Mathematics Extension 1 marking guidelines 2016

© NSW Department of Education, Sep-2320

Sample question 15i. When y=0

Vt sin θ−12g t 2=0

t ¿

t=2V sin θg as t>0

Range, xRange when t=2V sin θ

g seconds

xRange=V ( 2V sin θg )cosθ

¿ 2V 2 sin θ cosθg

¿ V2sin 2θg

as sin 2θ=2sin θ cosθ

ii. x.

=Vcosθ and y.

=Vsin θ−¿⇒ dydx

=Vsinθ−¿Vcosθ

Substitute θ=π3 and t=

2V√3 g

x.

=Vcos π3=V

2

y.

=Vsin π3−g( 2V

√3g )¿V . √3

2−2V

√3

¿−√36

V

tan α= y.

x. , tan α=

−√36

V

V2

, α=5π6

The projectile makes an angle 5π6 or 150 ° to the positive x-axis.

iii. The angle is obtuse, therefore it is descending.

32 ME-V1 Introduction to vectors

Source: Question 14a ©NSW Education Standards Authority HSC Mathematics Extension 1 marking guidelines 2015

© NSW Department of Education, Sep-2320

Sample question 16

i. Let AB=a and BC=b

AC=a+b and EF= 12a+ 1

2b=1

2 (a+b)=12AC, therefore EF ∥ AC

ii. Let AD=c and DC=d

AC=c+d and HG=12c+ 1

2d=1

2 (c+d)=12AC , therefore HG∥ AC and EF ∥ HG.

|EF|=|HG|, therefore EF=HG.

Therefore EFGH is a parallelogram

Sample question 17

i. AC .w=x2 y2− y2x2=0 therefore w is perpendicular to AC

ii. Scalar projection of AB on to w=AB .w

|w|=

x1 y2− y1 x2

√ x22+ y2

2

iii. Area of Triangle ABC=¿ 12×Scalar projection of AB on to w∨×|w|=1

2|x1 y2−x2 y1|

iv. PQ=(22) and PR=( 5−3)

Area ¿12|2× (−3 )−2×5|=8 units2

34 ME-V1 Introduction to vectors