o
~()hrp
fnr
-r-
.jf
()
«:
;yo
«:
0,."..'
A scalar is a quantity which has only size or magnitude.
A vector is a quantity with both magnitude and direction.
Two vectors are equal if they have the same magnitude
direction.
and
VECTORS IN GEOMETRIC FORM
Given two vectors v and w,
to find v + w we first start
by drawing v, and then from
the end of v we draw w.
The vector v + w starts at
the beginning of v and ends
at the end of w.
To find v - w we find
v + (-w). We start by
drawing v, then from the
end of v we draw =w.
'.I.I:~.'~
\-+--'
~~
~Vl
~V~~4--h-+~n='--rrr
i
'
~---+---
~I
t
__
The zero vector 0, is a vector of length O. It is the only vector
with no direction.
In the diagram alongside,
a+b=c
or a
+b
- c
= O.
VECTORS IN COMPONENT FORM
The basic unit vectors with magnitude 1 are:
in 2-D: i =
(~)
in 3-D: i =
(~).
j
The zero vector 0 is
(~)
and j
=
= (~)
(!).
and k
in 2-D and
= (~)-
(~)
in 3-D.
The general 2-D vector
v
= ( ~~) =
VIi
+ V2j
The general 3-D vector
v
= (~:)
VIi
+ V2j + v3k.
=
In examinations, scalars are written in italics, for example
and vectors are written in bold type, for example a. On paper
you should write vector a as
a:.
should understand the following for vectors in component
The scalar or dot product
v• w
•
vector equality
•
vector addition
•
a+b=b+a
(a+b)+c=a+(b+c)
•
•
a+O=O+a=a
a + (-a) = (-a)
vector subtraction
•
If v
•
•
•
if k
if k
> 0,
< 0,
ka and a have the same direction
•
•
if k
= 0,
ka
•
the distance between two points in space is the magnitude
of the vector which joins them
•
I ka'
•
k(a+b)=ka+kb
1
=
VV12
+
V22 + V32
the magnitude of vector v,
,k
v
and w
= (::)
then
+ V2W2 + V3W3.
VIWI
•
v e w = 0
•
v = kw
•
'v.
{=}
v is perpendicular
to w
v is parallel to w
{=}
w ] = [v
'I w]
v is parallel to w
{=}
If v • w
>
If v • w
<0
0 then () is acute.
then () is obtuse.
w
LIMES
11 a ,
The vector equation of a line is r = a + tb where a is the
position vector of any point on the line, b is a vector parallel to
the line, and t E JR.
POSITIOM VECTORS IM 2·D
The position vector of A(x,
y)
oA
is
or a
= (~)
For example, if an object has initial position vector a and moves
with constant velocity b, its position at time t is given by
r = a + tb for t ~ O.
(x, y, z)
or
+ yj.
xi
where () is the angle between the vectors.
point is given by cos () = IVy ~ :,.
0
•
=
= (~:)
e
The angle () between vectors v and w emanating from the same
ka and a have opposite directions
1
w 1 cos
For non-zero vectors v and w:
+ a = 0
a - b = a + (-b)
=
1 v 11
v ew =
by a scalar k to produce vector ka which is
multiplication
parallel to a
=
of two vectors
The position vector of B relative to A is
--+
--+
--+
=
OB - OA
Given
A(XI' YI)
AB
_' vector
•
the length of
•
th e
idooi
point
llll
.
POSITIOM
=
If r = (~).
b - a.
and B(x2,
AB
0
a = (~~).
and b = ( :: ).
//(
1:
the parametric form for the equation of a
line is x = Xo + tl, Y = Yo + tm,
Y2):
is AB = V(X2
-
XI)2
fAB--+'IS (XI+X2
Yl+Y2)
--2-' --2-
+ (Y2 - YI)Z
z
I )
= Zo + tn.
(Xo, Yo, zo)
.
The acute angle () between two lines is
given by cos () =
VECTORS IM 3-D
The position vector of A(x,
oA
Y, z) is
I,:~:,1
where a and
b are the direction vectors of the lines.
or a = (~)
or
+ yj + zk.
xi
Lines are:
--+
•
•
the length of AB is
AB
h
=
idnoi
V~(X-2---X"""'1)"""2-+-('--y-z---Y-I-:-;)
t e nu point 0
fAB'
IS
2'+----'( z-Z---Z"""'l
)"2
(XI+XZ
YI+Y2
--2--' --2-'
ZI+Z2)
--2-
PROPERTIES OF VECTORS
--+
D.
A, B, and Care collinear
parallel
•
coincident
•
intersecting if you can solve them simultaneously
a unique common point that fits both equations
•
skew if they are not parallel and do not have a point of
intersection; there are no solutions when the equations are
solved simultaneously.
.
if their direction vectors are parallel:
a
=
•
kb
if they are parallel and have a common point
to find
The shortest distance from point A to a line with direction vector
--+
if AB = kBC for some scalar k.
--->
b occurs at the point P on the line such that AP is perpendicular
to b.
Two vectors a and b are parallel if a = kb for some constant
#0.
The unit vector in the direction of a is
r;T1 a.
A vector b of length k in the same direction as a is b =
r;Tk a.
A vector b of length k which is parallel to a could be
k
=
a.
±r;T
III
Mathematics
SL - Exam Preparation
& Pradice
Guide (3rd edition)
SKILL BUILDER QUESTIONS (NO CALCULATORS)
BD in terms
a Write the diagonals AC and
-+
y
b
Calculate the vector dot product
and b.
of a and b.
-+
AC e BD in terms of a
-+
e Show that if ABCD i a rhombus then AC is perpendicular
-+
Q(2,3)
to BD.
P( -4,1)
q
7
Find t if
(~)
\!)
Consider
the points
~----------~~------------~x
a Write down the vectors p and q.
b The point R is defined by the vector
coordinates of R.
e
p + q.
Find the
a
-->
OA and b
A(l,
and
3)
~------~~--~-----.x
-+
Find the two points on (OM) which are 2VlO units from
M.
Consider vectors
a
=
3i - 6j and b
=
-+
®
Find the length of:
-+
ii AB.
Find the values of rand
Line L1 passes through the point
(-5,
and is parallel to
(~).
e Find the measure of OAB.
a Write a vector equation for the line.
b
Hence find the area of triangle OAB.
s.
-2)
d
+L
Find the point on the line with x-coordinate
e The line L2 is perpendicular
= (
7i + 2j.
C(5, 22) can be located using the vector
OC = ra + sb.
A(3, -1)
a Find AB.
a
Suppose
M.
-+
Consider the position vector
-1).
B(5,
-+
OB, and let M be the midpoint of AB.
e Write a vector equation for the line passing through 0 and
Y
The point
3
vectors.
b Hence find the coordinates of M.
9
iOA
are perpendicular
----t
d
b
=
(t~54)
a Write OM in terms of a and b.
Classify quadrilateral OPRQ.
B( -1,7)
=
and
to L1, and passes through
(4, 5).
~2 ).
Write a vector equation for the line L2.
ii Find the point of intersection of L1 and L2.
a Write a in terms of the base vectors i, j, and k.
b
iii Hence find the shortest distance from (4, 5) to the
line L1.
Find the magnitude of a.
e Write down a unit vector in the opposite direction to a.
"
Find the value(s) of
a parallel
k
for which
b
(~)
and
11
(:k)
are:
perpendicular.
b
§
B(-1,2)
The line in a makes
= 1 + kt,
y
=
i + 3j.
an angle
of
@ Triangle
-+
a Find BC.
------?
OA
Find the coordinates of D.
=
-i
( - 2, -1)
with
the line
2 - t, t E JR.
13k2
Show that k satisfies the equation
-
12k - 3 = O.
ABC is defined by the following information:
--+
-2i
+ 4j,
~
=
AB e OA
---t
0,
=
CA
-6i
+ 2j,
-+
e Find the angle between the diagonals [AC] and [BD].
and CB is parallel to j.
Hence classify quadrilateral ABCD.
a Find the coordinates of:
i A
B
6
=
Find the cosine of the angle between v and w.
x
d
5i - 2j and w
a Find the vector equation of the line through
which is parallel to the vector 3i + j.
b
b
=
a Find scalars rand s such that
r(v - w) = (r+s)i
- 20j.
__---,D
5
Vectors v and w are given by v
ii C
iii B
->
a
b
Find the point P on [BC] such that AP is perpendicular
to
-+
BC.
A
e
C
b
Parallelogram
-+
and AD
Mathematics
D
ABCD is formed by vectors
-+
AB
=
-+
DC
=
a,
b.
SL - Exam Preparation & Practice Guide (3rd edition)
Suppose
a
-+
= BC =
e Hence find the area of triangle ABC.
El
m
[m+n[
=
3i + j + 2k and n
b
=
-2i + 5j - 4k.
Find:
men
e the vector equation of the line passing through (1, -1, 2)
which is parallel to m.
-=-mdt given that
(~~)
tors.
(J )
and
7
are perpendicular
21
Suppose
a
= (~),
b
(i
=
8
b is perpendicular
to c - 2a.
1),
c
=
(T)'
and
Find the value of k.
lite vector equations for these lines:
@A
a parallel to (~)
is (-1,2,1)
a Write a vector equation for the line (AB) in the form
r = a + tb, t E R
and passing through the point (5, 0, -2)
b parallel to 2i - 3j
(-2,5,4)
+ 4k
and B is (0, 1,3).
b Find the angle between (AB) and the line L given by
and passing through the point
c perpendicular to the X 0 Z plane and passing through the
point
(2, -4,
1).
Find the parametric equations of these lines:
(i3)
a passing through (5, -1, 2) and parallel to
b passing through
(0, 2, -6)
c passing through
(2, 5, 1) and (8, 6, 2).
SKILL BUILDER QUESTIONS (CALCULATORS)
and parallel to 2i - 3j
Consider the points A(2, 10), B( -2, 2),
Suppose M is the midpoint of [BC). Find:
+k
--+
----t
a AC
bAM
and
C(16, 6).
c the size of angle BAC.
+ 5j,
2 For p = 4i - 9j and q = -12i
alql
find:
bpeq
c the angle between p and q
a Find x such that
a - 4x
=
d
2b.
a unit vector parallel to q.
y
3
b Find constants p and q if
(!3)
.
Q(4,10)
is parallel to a.
• R(10, 6)
c Find the possible values of k if
perpendicular
(
to b.
~~o )
IS
k+ 10
'-~B-------------------------~X
.
a Line Ll passes through A(l, -1,2)
P(6, -4)
and B(5, -1, -1).
Write a vector equation for the line.
a Show that
ii Find a point on L, which is 20 units from A.
m
b
C( 4, 1, -
¥)
c Show that 0, M, and R are collinear.
and is parallel to
o
to L2.
i State the y-coordinate
ine Ll passes through (6, 17) and is parallel to -i - 2j.
Line L2 passes through (0, 5) and (4, 2).
c L, meets L2 at the point P.
a Write a vector equation for L, in the form
of P.
b
ii Find the coordinates of P.
Find the shortest distance from C to the line L1.
-I2b.
Given a e b < 0, what conclusion can you draw about
the angle between a and b?
b
a Locate the train at time t = O.
a = ( ~)
and b = (~)
Consider vectors
a =
(Y)
and b = ( ~1 ).
i Find a e b.
ii Hence find the angle between
How long does it take for the train to reach B?
c Suppose
+ sb.
i Find a direction vector for L2.
ii Hence write a vector equation for L2.
After t minutes, the train is at point P with position vector
b
rr = a
c Find the acute angle between the lines L, and L2.
A mountain railway runs straight up a mountainside with the
aid of a cable. The train begins at point A with position vector
a, and ends at point B with position vector b.
(l--I2)a+
quadrilateral
b Find the midpoint M of [PQ].
Write a vector equation for the line.
p=
Hence classify
d Find a vector equation for the line (OR).
ii Show that Ll is perpendicular
d
--->
OP.
OPRQ.
At what point does Ll meet the YOZ plane?
Line L2 passes through
-3i + 2j - 4k.
----t
QR
these
degrees, correct to one decimal place.
where the units
are kilometres.
Find the distance between A and B.
ii Find the average speed of the train, giving your
answer exactly.
m
m
vectors
III
Find the area of the
triangle defined by a
and b.
Mathematics
SL - Exam Preparation
& Pradice
Guide
(3rd edition)
In triangle ABC, M is
the midpoint of [AB]
and N is the midpoint of
[AC].
A
N
b
e Show that the shortest distance
3y'33
.
from M to (KL) is
--u- units.
---+
Suppose
Write parametric equations for the line passing through K
and L.
AM = a and
-+
AN=b.
B<------------>.C
E)Lines
L
a Write in terms of a and b:
AB
i
AC
ii
iii
MN
I
BC
iv
b Explain why the line segment joining the midpoints of two
sides of a triangle is always parallel to the third side and
half its length.
7 Quadrilateral ABCD has vertices
C(1l,5),
and D( -1,2).
---+
Find the acute angle between the diagonals AC and BD.
(!) A
boat moves in a straight line defined by the parametric
equations
x = 3 + t and y = 4t - 3, where t is the
time in seconds, t;;, 0, and x and y are measured in metres.
a the initial position of the boat
Find:
b the position of the boat after 3 seconds
e the velocity vector of the boat
d
the speed of the boat.
0suppose
i represents a 1 km displacement
.
represents a 1 km displacement due north.
located at the point (0, 10). A ship is moving
with parametric equations x = 3 - 2t, Y =
where t is the number of hours after 8:30 am.
due east and j
A lighthouse IS
in a straight line
3t + 1, t > 0,
a What was the position of the ship at 8:30 am?
b Find the ship's:
ii speed.
i velocity vector
e Find the distance between the ship and the lighthouse at
10:30 am.
d
Find the time when the ship is directly
lighthouse.
west of the
e At what time (to the nearest minute) is the ship closest to
the lighthouse?
-,
Write vector equations in the form r = a + tb to dei le
the position at time t for these remote controlled toy rac. ng
cars. All distances are in metres and time is in seconds
Car A is initially at
y'i3
I
ms-
(9, -3),
in the direction
-2i
and is travelling
t
+ 3j.
ii Car B is initially at (-1, 4), and travels at a constant
velocity for 3 seconds to (5, 7).
b Show that the paths of the cars intersect at (3, 6).
e Will the cars collide? Explain your answer.
"
For the points
·find:
---+
a AB
A(2, 0, 5),
B(-3,
---+
1,7),
and
C(4, -2;9),
e the size of BAC.
b AC
12 A(2, -5, 3), B(7, -3, -1),
are vertices of a quadrilateral.
Ct l, 3, 0),
and
D(-4,1,4)
a Prove that ABCD is a parallelogram.
b
Find the size of the smaller angles of the parallelogram.
®consider
the points
M(3, -2, 5).
K(4, -2,
7), L(6, 1, -1),
and
a Find the measure of KLM.
Mathematics
SL - Exam Preparation
& Practice
Guide
(3rd edition)
:
~3
(~)
E
are defined by:
(
i1)
+t
L2:
x = 5 - r, y = -4
L3:
x = 5 - 38, y = 5
a Show that L1;,'
b Show that L; and
them.
A(2, 10), B(12, 8),
---+
L1, L2, and
..
L
J
( ;~)
+ 2r,
+ 48,
Z
Z
= 1+r
= 1 + 28
L2.
.ersect, and find the angle between