Maths Quest C Year 11 for Queensland
WorkSHEET 7.2
1
Chapter 7 Introduction to vectors WorkSHEET 7.2
Introduction to vectors
Express the vector shown below in terms of
i and j.
~
1
Name: _________________________
10 cos30 i +10 sin30 j
~
1
~
~
= 8.66 i + 5 j
~
2
Express the vector u~  2 ~i  3 j in terms of its
4
1
u  2 2  32
~
3
~
~
magnitude and the angle it makes with the
x-axis.
 13
3
tan  
2
  56
If u  i  3 j and v  2 i  j calculate u~ .v~.
u . v  (1  2  3  1)
~
~
~
~
~
~
If u~  ~i  2 j and v~  2 ~i  j
~
~
(a)
Show that these vectors are
perpendicular.
(b)
Find unit vectors in the v~ direction and in
~ ~
(a)
1
 1
u .v  2  2
3
~ ~
0
Therefore the vectors are perpendicular.
v^ 
(b)
~
a direction perpendicular to v~ .
^
~
v 
1
5
(2 i  j )
~
1 (i 
5 ~
~
2 j)
~
Maths Quest C Year 11 for Queensland
5
Chapter 7 Introduction to vectors WorkSHEET 7.2
Find value(s) of p, such that p ~i + 2 (1  3p) j
~
is perpendicular to 2p ~i + 3 j
Dot product should be zero.
2
3
(p  2p) + (2[1  3p]  3) = 0
~
2p2 + 6 18p = 0
p2  9p + 3 = 0
p
9  9 2  4(1)(3)
2
9  81  12

2
9  69

2
6
Find the equation of the path of the
time-varying position vector
1
u~ = ~i + 2 (t2  1) j .
~
t
State the type of path (linear, parabolic, and so
on). Hence, sketch its graph, and indicate the
direction of the path as t increases.
1
1
; t = ; y = 2(t2 – 1)
x
t
x=
y=2(
=
3
1
 1)
x2
2
2
x2
The path is hyperbolic.
7
For a particle which has a position vector u~
given in the previous question, how far is it
from the starting point after it has been
travelling for 5 seconds?
8
When t = 5, u~ =
~
 48
~
What is the equation of this path
What is the period of this motion?
1
u  0.04  48 2
Consider the particle which moves according to (a)
the equation: u~  2 cos3t ~i  2 sin3 t j .
(a)
(b)
1
i + 48 j .
~
5 ~
(b)
Circle, radius 2
x2 + y2 = 4
2
The period is
3
2
Maths Quest C Year 11 for Queensland
9
Chapter 7 Introduction to vectors WorkSHEET 7.2
For the vectors
u~  ~i  2 j  k~ and v~  2 ~i  j  k~ , find the
~
~
scalar resolute of u~ on v~
 2i  j  k
~

~
~
u~ . v~   ~i  2 j  k~ .
~
2

  2   1  12

2  2 1

6


10
Using the same vectors as in the previous
question, find:
(a) the vector resolute of v~ on u~ .
(b)
the vector resolute of
v~ perpendicu lar to u~ .
(a)
1
6
or
3




3
6
6
4
 
v   u~ . v~  u~
11


 i  2 j  k 
 i  2 j  k
~
~
~
~




~
~

 
. 2 ~i  j  k~ 

~
 12  2 2  12  
 12  2 2  12



 2  2  1  ~i  2~j  k~
 

6 
6

1
1 



 ~i  2~j  k~ 

6
6
1

  ~i  2 j  k~ 
~
6

(b)
V~  V~  V~

11
1
 2~i  j  k~   ~i  2 j  k~ 
~
~
6

1
1


 12~i  6 j  6k~    ~i  2 j  k~ 
~
~
6
 6

1

 11 ~i  8 j  5k~ 
~
6

~



