11SM Vectors I Test A · 2019-10-19 · ©John Wiley & Sons Australia, Ltd 8 Chapter 2: Vectors in the plane Test A Name: _____ Complex unfamiliar 17 Calculate the equation of the

© John Wiley & Sons Australia, Ltd 1

Chapter 2: Vectors in the plane Test A Name: _____________________ Simple familiar 1 If = 4 - 3 and = 2 + 4 , then

determine the vector 4 - 2.5 .

[1 mark]

[1 mark]

2

2 Which of the following vectors has a magnitude of 6? A 5 + 5 B 2 - 5 C 5 - 2 D 36

D 1

3 Evaluate the magnitude of the vector PQ if the position vector for point P is 3 + 4 and for

point Q is - + 3 .

[1 mark]

[1 mark] ** the above is the textbook solution. Can you see their stuffing around has caused an error! Draw P and Draw Q … you can CLEARLY see the 𝑃𝑄##### = 𝑃# − 𝑄# If you can’t see this, you need to see Mr Finney to show you! Now;

𝑃𝑄##### = 𝑝) − 𝑞) = 3𝑖 + 4𝑗 − (−𝑖 + 3𝑗) = 3𝑖 + 4𝑗 + 𝑖 − 3𝑗

= 4𝑖 + 𝑗 And

2𝑃𝑄##### 2 = 344 + 14 = √5

What a great learning opportunity. Draw the vectors to evaluate that you are correct and the textbook is wrong!

2

~u i j ~

v i j

~u

~v

( ) ( )2

ˆ ˆ ˆ ˆ4 24 3 2.5 4

ˆ ˆ ˆ ˆ1

1 5 016

i j i j

i j i j= - -

- +-

-ˆ11ˆ 22i j-=

i ji ji ji

i~j

i j

ˆ ˆ ˆ ˆ(3 4 ) ( 3 )ˆ ˆ4 7

PQ PO OQ

i j i ji j

= +

= - + + - +

= - +

16 49

65

PQ = +

=

Maths Quest 11 Specialist Mathematics Units 1 & 2 for Queensland Chapter 2: Vectors in the plane Test A


4 If = 3 + 4 and v = -4 + 5 , then

calculate the value of .

1

5 Evaluate the size of angle between = 3 + 4 and = -4 + 5 ,

correct to 2 decimal places.

[1 mark]

[1 mark]

2

6 Determine a unit vector perpendicular to -3 + 4 .

When two vectors are perpendicular, their dot product is zero. [1 mark] A vector perpendicular to is . A unit vector perpendicular to is

. [1 mark]

2

7 If = -3 + a , = 2a + a and is

perpendicular to , then determine the value of

a.

[1 mark] If a = 0, then the vectors are equal to zero, so a = 6. [1 mark]

2

~u i j i j

~u v× !

3 4 4 512 208

u v× = ´- + ´= - +=

! !

~u i j ~

v i jcos

8u v u vu v

q× =

× =! ! ! !

! !9 165

u = +

=!

16 25

41

v = +

=!

1

8cos5 41

8cos5 41

75.53

q

q

q

-

=

æ ö= ç ÷è ø

= °

i jˆ ˆ3 4i j- + ˆ ˆ4 3i j+

ˆ ˆ3 4i j- +1 ˆ ˆ(4 3 )5i j+

~u i j ~

v i j~u

~v

23 2u v a a× = - ´ +!" 0u v× =!" 2

2

3 2 06 0

( 6) 00 or 6

a aa aa a

a a

- ´ + =

- =- =

= =



8 Calculate the size of the angle between = 5 - 2 and = 3 + 4 , leaving your

answer as an exact value.

[1 mark]

[1 mark]

2

9 Calculate the scalar resolute of on if

= + 2 and = 2 + 3 .

Calculate the magnitude of

[1 mark]

[1 mark]

2

10 Referring to vectors and in question 9,

determine the vector resolute of parallel to

.

[1 mark]

[1 mark]

2

11 Let (t) be a position vector of an object

whose position varies with time. If = 3 sin t + 3 cos t then the path this

object takes is: A a straight line B a parabola C an ellipse D a circle

[1 mark] The path the object takes is a circle. D [1 mark]

2

~u i j ~

v i jcosu v u v q× =

! ! ! !7u v× =! !

25 4

29

u = +

=!

9 165

v = +

=!

1

7cos5 29

7cos5 29

q

q -

=

æ ö= ç ÷è ø

~a ~b

~a i j ~b i j

b!ˆ

1 ˆ ˆ(2 3 )13

bbb

i j

=

= +

!"

!

1ˆ ˆ ˆ ˆ ˆ(2 3 ).( 2 )131 (2 1 3 2)13813

b a i j i j× = + +

= ´ + ´

=

!"

~a ~b

~a

~b

ˆ ˆ( )8 1 ˆ ˆ(2 3 )13 13

a b a b

i j

= ×

= × +

! " "# #

8 ˆ ˆ(2 3 )13

i j= +

~u

~u i ˆ,j

3sin , 3cosx t y t= =2 2 2 29sin , 9cosx t y t= =2 2 9x y+ =



12 Let (t) be a position vector of an object

whose position varies with time. If = 2t + (t2 - t) then the path this object

takes is: A a straight line B a parabola C an ellipse D a circle

[1 mark]

The path the object takes is a parabola. B [1 mark]

2

~u

~u i ˆ,j

22xx t t= Þ =

2

2

2

2 21 14 2

y t t

x xy

y x x

= -

æ ö= -ç ÷è ø

= -


Chapter 2: Vectors in the plane Test A Name: _____________________ Complex familiar 13 If and , determine if:

(a) and are parallel (b) and are perpendicular (c) and are equal in length

(d) the scalar resolute of on is .

(a)

[1 mark]

(b)

[1 mark]

(c)

[1 mark]

(d)

[1 mark]

4 ˆ ˆ3 6a i j= -!

ˆ ˆ3b pi j= +!

pa! b

!a! b!a! b!

a! b!

55

a kb=! !ˆ ˆ ˆ ˆ3 6 3

3 ˆ ˆ ˆ ˆ2( 3 ) 32

32

i j pi j

i j pi j

p

- = +

- - + = +

= -

0a b× =! !ˆ ˆ ˆ ˆ(3 6 ) ( 3 ) 0

3 18 03 19

6

i j pi jp

pp

- × + =- =

==

2 23 ( 6)

9 36

45

a = + -

= +

=

!

2 2

2

(3)

9

b p

p

= +

= +!

2

2

2

45 9

45 945 96

a b

p

ppp

=

= +

= +

= -=

! !

5ˆ5

aa b ba

× = × =!! ! !

!ˆ ˆ5 3 6 ˆ ˆ( 3 )5 45

ˆ ˆ5 3 6 ˆ ˆ( 3 )5 3 53 3 1853 15 90

15 93165

i j pi j

i j pi j

p

pp

p

-= × +

-= × +

= -

= -=

=



14 Let = 5 + 3 and = + 2 .

(a) Calculate the angle between the two vectors, in radians, to 4 decimal places.

(b) Find , the angle which makes with the x-axis, in radians rounded to 4 decimal places.

(c) Find the vector resolute of in the direction of .

(a)

[1 mark]

(b)

[1 mark]

(c)

[1 mark]

3 ~u i j ~v i j

a u!

v!u

!

2 2 2 2

1

5 1 3 2cos5 3 1 21134 50.8437

cos 0.84370.5667

q

q -

´ + ´=

+ +

=´

=

==

1 3 tan5

0.5404

a - æ ö= ç ÷è ø

=

2 25 3

366

u = +

==

!

ˆ

1 ˆ ˆ(5 3 )6

uuu

i j

=

= +

!!!

1 ˆ ˆ ˆ ˆˆ (5 3 ) ( 2 )61 (5 6)6116

u v i j i j× = + × +

= +

=

! !

ˆ ˆ( )

11 1 ˆ ˆ(5 3 )6 655 33ˆ ˆ36 3655 11ˆ ˆ36 12

v u v u

i j

i j

i j

= ×

é ù= +ê úë û

= +

= +

!" " "



15 Let = 3 -5 and = -4 + . Evaluate:

(a) +

(b) -

(c)

(d)

(e) the angle between and .

(a) + = (3 - 4) + (-5 + 1) [1 mark]

= - - 4 [1 mark] (b) - = [3 - (-4)] + (-5 - 1)

[1 mark] = 7 - 6 [1 mark]

(c) = (3 ´ -4) + (-5 ´ 1) [1 mark]

= -12 - 5 = -17 [1 mark]

(d) = [1 mark]

=

= (3 - 5 ) [1 mark]

(e) The angle between and [1 mark]

[1 mark]

10

16 Let = 2 + 4 . Determine a vector, parallel

to , such that their dot product is 40.

m = m(2 + 4 )

m = m(2 + 4 ) (2 + 4 ) = 40

[1 mark] 4m + 16m = 40 20m = 40 m = 2 [1 mark] The new vector is 4 + 8 . [1 mark]

3

~U i j ~v i j

~U ~v

~U ~v

~U × ~v

^

~U

~U ~v

~U ~v i j

i j

~U ~v i j

i j

~U × ~v

^

~U~

~

UU

2 2

ˆ ˆ3 53 5i j-+

341 i j

~U ~v

1

1

17cos34 171cos2

135

-

-

-æ ö= ç ÷è ø-æ ö= ç ÷

è ø= °

~u i j

~u~u i j

~v× ~u i j × i j

i j


Chapter 2: Vectors in the plane Test A Name: _____________________ Complex unfamiliar 17 Calculate the equation of the path of the time-

varying position vector = + 2 (t2 - 1) .

State the type of path (linear, parabolic and so on). Hence, sketch its graph, and indicate the direction of the path as t increases.

x = so t =

y = 2(t2 – 1) [1 mark]

y = 2

= - 2 [1 mark]

The path is hyperbolic. [1 mark]

[1 mark for curve, 1 mark for direction]

5

~u t1 i j t

1x1

÷øö

çèæ -11

2x

2

2x

y = –2



18 A trekker walks 5 km due west from H, then turns northward at a bearing of 060° (N60°E) for a distance of 10 km, and then travels due north for a further 5 km to point X. (a) Determine the position vector from H to

X. (b) Calculate the distance from H to X

(correct to 2 decimal places).

Path taken by trekker:

(a) Add up all the vectors.

= (-5 + 10 sin 60° + 0) + (0 + 10 cos 60° + 5) [1 mark] = 3.66 + 10 [1 mark]

(b) Distance HX

(correct to 2 decimal places) [1 mark]

3

19 The top of a 10-m diving board lies over the swimming pool as illustrated below.

Sally sits 30 m away in the corner of the swimming pool and takes a bearing of 40° (N40°E) of the feet of her friend who is about to do a belly flop. State the position vector from Sally’s current position to her friend’s feet.

Assume that Sally’s position is the origin. Then from the origin, Sally’s friend’s feet will have the following coordinates: the x-coordinate will be 30 sin 40°, [1 mark] the y-coordinate will be 30 cos 40° [1 mark] and the z-coordinate will be 10. [1 mark] The position vector is: 30 sin 40° + 30 cos 40° + 10 = 19.28 + 22.98 + 10 . [1 mark]

4

HX!!!"

ij

i j

2 2

HX

3.66 10

113.395610.65km

=

= +

==

i j ki j k



20 Show that vectors and are parallel using the dot

product rule. (Note: Marks will not be awarded if the dot product rule is not used.)

If two vectors are parallel, the angle between them is zero. [1 mark]

[1 mark]

[1 mark]

[1 mark]

4

2ˆ ˆv mi m j= +!2 3ˆ ˆw m i m j= +!

1 2 1 2cos x x y yv w

q +=

! !LHS cos01

==

2 2 3

2 2 2 2 2 3 2

3 5

2 4 4 6

3 2

2 2 4 2

3 2

2 2 2

3 2

3 2

RHS( ) ( ) ( )

(1 )(1 ) (1 )

(1 )(1 ) (1 )

(1 )(1 )

1LHS RHS

m m m mm m m m

m mm m m m

m mm m m m

m mm m m m

m mm m

´ + ´=

+ ´ +

+=

+ ´ ++

=+ ´ +

+=

+ ´ +

+=

+==

Documents

11SM Vectors I Test A · 2019-10-19 · ©John Wiley & Sons Australia, Ltd 8 Chapter 2: Vectors in the plane Test A Name: _____ Complex unfamiliar 17 Calculate the equation of the