IB Standard Level
Vector Review
1.
Name:
The vertices of the triangle PQR are defined by the position vectors
 4 
 6
3
 
 
 
OP    3  , OQ    1 and OR    1 . Find
(adapted 2009 p1)
 1 
 5
2
 
 
 
a) PQ
[2 marks]
b) PR
[1 mark]
1
c) Show that cos RPˆ Q  .
2
[7 marks]
d) Find sin RPˆ Q .
[3 marks]
e) Hence, find the area of triangle PQR, giving your answer in the form a 3 .
[3marks]
 2  5 
 9    3
   
   
2. Two lines with equations r1   3   s  3  and r2   2   t  5  intersect at the point P.
  1  2 
 2   1
   
   
Find the coordinates of P.
(adapted 2009 P2)
[6 marks]
3. The diagram shows a cube, OABCDEFG where the length of each edge is 5cm. Express the
following vectors in terms of i, j, and k.
(adapted 2006)
a) OG
[2 marks]
b) BD
[2 marks]
c) EB
[2 marks]
4. A triangle has its vertices at A(-1, 3), B(3, 6) and C(-4, 4).
(adapted 2006)
a) Show that AB  AC  9 .
[3 marks]
b) Show that, to three significant figures, cos BAˆ C  0.569
[3 marks]
5. In this question, distance is in kilometers, time is in hours.
A balloon is moving at a constant height with a speed of 18kmh1 , in the direction of the
3
 
vector  4  . At time t = 0, the balloon is at point B with coordinates (0, 0, 5). (adapted 2005 P2)
0
 
 x   0  10.8 
    

a) Show that the position vector b of the balloon at time t is given by b   y    0   t 14.4  .
 z  5  0 
    

[6 marks]
At time t = 0, a helicopter goes to deliver a message to the balloon. The position vector h of the
 x   49    48 
    

helicopter at time t is given by h   y    32   t   24  .
z  0   6 
    

b) Write down the coordinates of the starting position of the helicopter.
[2 marks]
c) Find the speed of the helicopter.
[3 marks]
The helicopter reaches the balloon at point R.
d) Find the time the helicopter takes to reach the balloon.
[3 marks]
e) Find the coordinates of R.
[2 marks]
6. Consider the points A(5, 8), B(3, 5) and C(8, 6). Find
(adapted 2008 P1)
a) AB
[2 marks]
b) AC
[1 mark]
c) Find AB  AC .
[2 marks]
d) Find the sine of the angle between AB and AC.
[1 mark]
 1    6
 9    2
   
   
7. Two lines L1 and L2 are given by r1   4   s 6  and r2   20   t  10  .
 2    2
  6   10 
   
   
(adapted 2008 P1)
52
a) Let θ be the acute angle between L1 and L2. Show that cos  
.
[5 marks]
140
b) P is the point on L1 when s = 1. Find the position vector of P.
[3 marks]
c) Show that P is also L2.
[5 marks]
 6 


d) A third line L3 has direction vector  x  . If L1 and L3 are parallel, find the value of x
  30 


[3 marks]
8. Let v  3i  4 j  k and w  i  2 j  3k . The vector v + pw is perpendicular to w. Find the
value of p.
(adapted 2008 P2)
[7 marks]
9. The point O has coordinates (0, 0, 0), point A has coordinates (1, -2, 3) and point B has
coordinates (-3, 4, 2).
(adapted 2008 P2)
  4
 
a) Show that AB   6  .
[3 marks]
 1
 
b) Find BAˆ O .
[4 marks]
 x    3   4 
     
c) The line L1 has equation  y    4   s 6  . Write down the coordinates of two points on
 z   2   1
     
L1.
[2 marks]
The line L2 passes through A and is parallel to OB .
d) Find a vector equation for L2, giving your answer in the form r  a  tb .
[2 marks]
e) Point C(k, -k, 5) is on L2. Find the coordinates of C.
[5 marks]
 x  3 
   
f) The line L3 has equation  y     8  
z  0 
   
of p at C.
 1 
 
p  2  , and passes through the point C. Find the value
 1
 
[2 marks]
k 
1
10. Two vectors are given by p    and q    , k  R .
(adapted 1995)
1
k 
a) Find the value of k for which p and q are mutually perpendicular.
[2 marks]
b) Find the two values of k for which the angle between p and q is 60º.
[3 marks]
1
 6 
11. Find the size of the angle between the two vectors   and   . Give your answer to the
  8
 2
nearest degree.
(adapted 2000)
[4 marks]
 2
12. A line passes through the point (4, -1) and its direction is perpendicular to the vector   .
 3
Find the equation of the line in the form ax  by  p , where a, b, and p are integers to be
determined.
(adapted 2000)
[4 marks]
 x   1   2
13. A vector equation of a line is       t   , t  R . Find the equation of this line in the
 y   2  3 
form ax  by  c , where a, b, and c   .
(adapted 2002)
[4 marks]
 5
 3 
  2  4
14. The vector equations of two lines are given by r1        and r2     t   . The
1
  2
 2  1
lines intersect at the point P. Find the position vector of P. (adapted 2003)
[4 marks]
15. Consider the vectors c  3i  4 j and d  5i  12 j .
(adapted 2003)
a) Calculate the scalar product c  d .
[3 marks]
b) Calculate the scalar projection of the vector c in the direction of the vector d.
[3 marks]
7
10 
16. The diagram shows a parallelogram OPQR in which OP    , OQ    .
1
 3
a) Find the vector OR .
[3 marks]
15
b) Use the scalar product of two vectors to show that cos OPˆ Q  
.
754
[4 marks]
c) Explain why cos PQˆ R   cos OPˆ Q .
[1 mark]
23
d) Hence show that sin PQˆ R 
.
754
[2 marks]
e) Calculate the area of the parallelogram OPQR, giving your answer as an integer. [4 marks]
 3 


17. The line L1 is represented by the vector equation r    1  
  25 


parallel to L1 and passes through the point B(-8, -5, 25).
 2 
 
p 1  . A second line L2 is
  8
 
(adapted 2010 P1)
a) Write down a vector equation for L2 in the form r  a  tb .
[2 marks]
 5   7
   
A third line L3 is perpendicular to L1 and is represented by r   0   q  2 
 3  k 
   
b) Show that k = -2.
[5 marks]
The lines L1 and L3 intersect at the point A.
c) Find the coordinates of A.
[6 marks]
 6 


The lines L2 and L3 intersect at point C where BC   3  .
  24 


d) Find AB .
[2 marks]
e) Find AC .
[3 marks]
−2
1
18. A line L passes through A(1, -1, 2) and is parallel to the line 𝒓 = ( 1 ) + 𝑠 ( 3 ).
5
−2
(adapted from IB 2011 p1)
(a) Write down a vector equation for L in the form r = a+tb.
[2 marks]
The line L passes through point P when t = 2.
(b) Find
⃗⃗⃗⃗⃗ ;
(i) 𝑂𝑃
⃗⃗⃗⃗⃗ |.
(ii) |𝑂𝑃
[4 marks]
−3
19. The following diagram shows the obtuse-angled triangle ABC such that ⃗⃗⃗⃗⃗
𝐴𝐵 = ( 0 ) and
−4
−2
⃗⃗⃗⃗⃗
𝐴𝐶 + ( 2 ).
(adapted from IB 2011 p1)
−6
⃗⃗⃗⃗⃗ .
(a) (i) Write down 𝐵𝐴
⃗⃗⃗⃗⃗ .
(ii) Find 𝐵𝐶
[3 marks]
̂ C.
(b) (i) Find cos AB
̂ C.
(ii) Hence, find sin AB
−4
⃗⃗⃗⃗⃗ = ( 5 ), where 𝑝 > 0.
The point D is such that 𝐶𝐷
𝑝
⃗⃗⃗⃗⃗ | = √50, show that p = 3.
(c) (i) Given that |𝐶𝐷
[7 marks]
⃗⃗⃗⃗⃗ .
(ii) Hence, show that ⃗⃗⃗⃗⃗
𝐶𝐷 is perpendicular to 𝐵𝐶
20. A line L1 passes through points P(-1, 6, -1) and Q(0, 4,1).
[6 marks]
(adapted from IB 2012 p1)
1
⃗⃗⃗⃗⃗
(a) (i) Show that 𝑃𝑄 = (−2).
2
(ii) Hence, write down an equation for L1 in the form r = a+tb.
[3 marks]
4
3
⃗ = ( 2 ) + 𝑠 ( 0 ).
A second line L2 has equation 𝒓
−1
−4
⃗⃗⃗⃗⃗ and L2.
(b) Find the cosine of the angle between 𝑃𝑄
[7 marks]
(c) The lines L1and L2 intersect at the point R. Find the coordinates of R.
[7 marks]
21. The following diagram shows two ships A and B. At noon, ship A was 15 km due north of
ship B. Ship A was moving south at 15 km h-1 and ship B was moving east at 11 km h-1.
(adapted from IB 2012 p2)
(a) Find the distance between the ships
(i) at 13:00;
(ii) at 14:00.
Let s(t) be the distance between the ships t hours after noon, for 0 ≤ t ≤ 4.
[5 marks]
(b) Show that 𝑠(𝑡) = √346𝑡 2 − 450𝑡 + 225.
[6 marks]
(c) Sketch the graph of s(t).
[3 marks]
(d) Due to poor weather, the captain of ship A can only see another ship if they are less
than 8 km apart. Explain why the captain cannot see ship B between noon and 16:00.
[3 marks]
2
⃗ = (1).
) 𝑎𝑛𝑑 𝒃
−3
4
⃗ =(
22. Consider the vectors 𝒂
(adapted from IB 2013 p1)
(a) Find
(i) 2a+b;
(ii) |2𝐚 + 𝐛|.
[4 marks]
Let 2a+b+c = 0, where 0 is the zero vector.
(b) Find c.
[2 marks]
23. Consider points A(1, -2, -1), B(7, -4, 3) and C(1, -2, 3). The line L1 passes through C and is
⃗⃗⃗⃗⃗ .
parallel to 𝐴𝐵
(adapted from IB 2013 p1)
(a) (i) Find ⃗⃗⃗⃗⃗
𝐴𝐵 .
(ii) Hence, write down a vector equation for L1.
[4 marks]
3
−1
⃗ = ( 2 ) + 𝑠 (−3).
A second line, L2, is given by 𝒓
𝑝
15
(b) Given that L1 is perpendicular to L2, show that p = -6.
[3 marks]
(c) The line L1 intersects the line L2 at point Q. Find the x-coordinate of Q. [7 marks]