IB Math SL – year 2
Past paper questions – Vectors (question set #1)
1.
 3
  2
Find the cosine of the angle between the two vectors   and   .
 4
 1
2.
The vectors i , j are unit vectors along the x-axis and y-axis respectively.
(Total 6 marks)
 







The vectors u = – i + 2 j and v = 3 i + 5 j are given.
(a)


Find u + 2 v in terms of i and j .



A vector w has the same direction as u + 2 v , and has a magnitude of 26.
(b)
3.



Find w in terms of i and j .
(Total 4 marks)
The following diagram shows the point O with coordinates (0, 0), the point A with
position vector a = 12i + 5j, and the point B with position vector b = 6i + 8j. The
angle between (OA) and (OB) is .
y
C
B
Find
4.
(i)
| a |;
(ii)
a unit vector in the direction of b;
(iii)
the exact value of cos in the form
A
p
, where, p, q  .
q
x
O
(6 marks)
 6
The circle shown has centre O and radius 6. OA is the vector   , OB is the
0
y
 
  6
 5 
 and OC is the vector, 
 .
 0 
 11 
vector 
C
B
O
(a)
Verify that A, B and C lie on the circle.
(3)
(b)
Find the vector AC .
(2)
(c)
Using an appropriate scalar product, or otherwise, find the cosine of angle OAC.
(3)
(d)
Find the area of triangle ABC, giving your answer in the form a 11 , where a 
. (4)
A
(Total 12 marks)
5.
The quadrilateral OABC has vertices with coordinates O(0, 0) A(5, 1) B(10, 5) and C(2, 7).
(a)
Find the vectors OB and AC .
(b)
Find the angle between the diagonals of the quadrilateral OABC.
(Total 4 marks)
1
x
7.
 – 30 
 60 
 and 
 .
 40 
 25 
6.
(a) Find the scalar product of the vectors 
(b)
Two markers are at the points P (60, 25) and Q (–30, 40). A surveyor stands at O (0, 0) and looks at
marker P. Find the angle she turns through to look at marker Q.
(Total 6 marks)
y
The triangle ABC is defined by the following information:
4
 2
OA =  
  3
 3
AB =   ,
 4
AB BC = 0
 0
AC , is parallel to   .
1
(a)
3
2
1
–2
On the grid at right, draw an accurate diagram of triangle ABC.
–1 O
–1
1
2
3
4
5
–2
(b)
Write down the vector OC .
–3
–4
(Total 4 marks)
8.
Find the angle between the following vectors a and b, giving your answer to the nearest degree.
a = –4i – 2j
9.
b = i – 7j
(Total 4 marks)
 2 
  3
 and   respectively.
  2
1
The points A and B have the position vectors 
(a)
The angle
Find the vector AB .
(i)
Find AB .
(ii)
(4)
(ii)
d

 23 
(b)
Find the vector AD
in terms of d. (2)
10.
BÂD is 90°.
(c) (i)
The point D has position vector 
(Total 15 marks)
Show that d = 7.
Write down the position vector of
the point D.
(3)
The quadrilateral ABCD is a rectangle.
(d)
Find the position vector of the point C.
(4)
(e)
Find the area of the rectangle ABCD.
(2)
 2x 
 x  1
 and 
 are perpendicular for two values of x.
 x – 3
 5 
The vectors 
(a)
Write down the quadratic equation which the two values of x must satisfy.
(b)
Find the two values of x.
(Total 4 marks)
2
6
x