Higher Outcome 4

1
3D Coordinates
In the real world points in space can be located
using a 3D coordinate system.
For example, air traffic controllers find the
location a plane by its height and grid reference.
z
(x, y, z)
y
x
2
Write down the coordinates for the vertices
(6, 1, 2)
(0, 1, 2)
(6, 0, 2)
(0, 0, 2)
(0, 1, 0 )
(6, 1, 0)
(0,0, 0)
(6, 0, 0)
3
3D vectors are defined by 3 components.
For example, the velocity of an aircraft taking
off can be illustrated by the vector v.
z
(7, 3, 2)
2
v
y
2
3
x
7
4
Any vector can be represented in terms of the
i , j and k Where i, j and k are unit
vectors (one unit long) in the x, y and z
directions.
z
y
j
k
x
i
5
Any vector can be represented in terms of the
i , j and k Where i, j and k are unit
vectors in the x, y and z directions.
z
(7, 3, 2)
v
y
2
v ( 7i 3j 2k )
3
x
7
6
Magnitude of a Vector
A vectors magnitude (length) is represented by
v
A 3D vectors magnitude is calculated using
Pythagoras Theorem in 3D
(3 , 2 , 1)
z
v
y
1
2
x
3
7
Find the unit vector in the direction of u
8
Addition of vectors
9
Addition of Vectors
For vectors u and v
10
Negative vector
For any vector u
11
Subtraction of vectors
12
Subtraction of Vectors
For vectors u and v
13
Multiplication by a scalar ( a number)
Hence if u kv then u is parallel to v
Conversely if u is parallel to v then u kv
14
CombiningVectors
15
Show that the two vectors are parallel.
If z kw then z is parallel to w
16
Position Vectors
A (3,2,1)
z
a
y
1
2
x
3
17
Position Vectors
If R is ( 2 , -5 , 1) and S is (4 , 1 , -3)
18
Position Vectors as a journey
If R is ( 2 , -5 , 1) and S is (4 , 1 , -3)
19
Vectors as a journey
Express VT in terms of f, g and h
h
h
f
g
f
g
20
Collinear Points
A is (0 , -3 , 5), B is (7 , -6 , 9) and C is (21
, -12 , 17). Show that A , B and C are collinear
stating the ratio ABBC.
AB and BC are parallel (multiples of each other)
through the common point B, and so must be
collinear
A
C
B
1
2
AB BC 1 2
21
Collinear Points
Given that the points S(4, 5, 1), T(16, 4, 16)
and U(24, 10, 26) are collinear,
calculate the ratio in which T divides SU.
S
U
3
2
T
ST TU 3 2
22
Using Vectors as journeys
PQRS is a parallelogram with P(3 , 4 , 0), Q(7 ,
6 , -3) and R(8 , 5 , 2). Find the coordinates of
S.
Q(7 , 6 , -3)
P(3 , 4 , 0)
Draw a sketch
The journey from Q to R is the same as the
journey from P to S
S
R(8 , 5 , 2)
S(3 1 , 4 1 , 0 5)
From P to S
S(4 , 3 , 5)
23
The point Q divides the line joining P(1, 1, 0)
to R(5, 2 3) in the ratio 21. Find the
co-ordinates of Q.
The journey from P to Q
Q(-1 4, -1 2, 0 2)
Q(3 , 1 , 2)
24
P divides the line joining S(1,0,2) and T(5,4,10)
in the ratio 13. Find the coordinates of P.
The journey from S to P
P(1 1, 0 1, 2 2)
P(2 , 1 , 4)
25
The scalar product
The scalar product is defined as being
a . b a b cos ?
0 ? 180
a
?
a and b must be divergent, ie joined tail to tail
b
26
The Scalar Product
Find the scalar product for a and b when a 4 ,
b 5 when (a) ? 45o (b) ? 90o
a . b a b cos ?
? a . b 4 5 cos 45o
? a . b 20 1/v2
v2/v2
? a . b 10v2
a . b a b cos ?
? a . b 4 5 cos 90o
? a . b 20 0
? a . b 0
When ? 90o
27
The Scalar Product
This equilateral triangle has sides of 3 units. p
. q
p . q p q cos ?
? p . q 3 3 cos 60o
? p . q 9 1/2
? p . q 41/2
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The Scalar Product
This equilateral triangle has sides of 3 units. p
. (q r)
r
r
p . (q r) p . q p . r
60o
? p . q 3 3 cos 60o
p and r are not divergent so move r
? p . q 3 3 ½
? p . r 3 3 cos 60o
? p . q 41/2
? p . r 9 ? ½
p . (q r) p . q p . r 9
? p . r 4½
29
If a and b are perpendicular then a . b 0
30
Component Form Scalar Product
a . b a1b1 a2b2 a3b3
31
Angle between Vectors
To find the angle between two vectors we simply
use the scalar product formulae rearranged
a . b a1b1 a2b2 a3b3
a . b a b cos ?
cos ?
cos ?
32
Find the angle between the two vectors below.
p 3i 2j 5k and q 4i j 3k
q v(42 12 32)
p v(32 22 52)
q v26
p v38
a . b a1b1 a2b2 a3b3
cos ?
34 21 53
0923
29
? cos-1 0923 227o
33
Perpendicular Vectors
a . b a b cosO
If a and b are perpendicular then a . b 0
cos 90o 0
If a and b are perpendicular then
a1b1 a2b2 a3b3 0
34
If a and b are perpendicular then a . b 0
? a . b 31 22 (-1)7
a . b a1b1 a2b2 a3b3
? a . b 3 4 7
? a . b 0
? a and b are perpendicular
35
Properties of a Scalar Product
Two properties that you need to be aware of
a . b b . a
a .( b c) a . b a . c
36
If p 5 and q 4, find p . (p q)
p
60o
q
p . (p q) p . p p . q
p p cos 0o p q cos60o
5 5 1 5 4 ½
25 10
35
37
P(-2,-1,-4)
T divides PR in the ratio 54
Show that Q, T and S are collinear, and find the
ratio in which T divides QS
S(7,2,17)
Q(1,5,-7)
Find the acute angle between the diagonals of PQRS
R(7,8,5)
?T(-25,-15,-45)
?T(3 , 4 , 1)
QT TS 1 2
38
P(-2,-1,-4)
Find the acute angle between the diagonals of PQRS
S(7,2,17)
Q(1,5,-7)
R(7,8,5)
39
Vectors u and v are defined by u 3i 2j
and v 2i 3j 4k Determine whether or not u
and v are perpendicular.
? u . v 32 2(-3) 04
? u . v 6 6 0
? u . v 0
Hence vectors are perpendicular
40
For what value of t are the vectors u and v
perpendicular ?
u . v 0 if vectors perpendicular
? u . v t 2 (-2)10 3t
? u . v 5t 20
u . v 0 ? 5t 20 0
t 4
41
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42
The diagram shows two vectors a and b, with
a 3 and b 2?2. These vectors are
inclined at an angle of 45 to each other. a)
Evaluate i) a.a ii) b.b iii)
a.b b) Another vector p is defined by p
2a 3b Evaluate p.p and hence write
down p .
ii)
i)
iii)
b)
p2
43
Vectors p, q and r are defined by p i j
k, q i 4k, r 4i 3j a) Express p q
2r in component form b) Calculate p.r c) Find
r
a)
b)
c)
44
The diagram shows a point P with co-ordinates (4,
2, 6) and two points S and T which lie on the
x-axis. If P is 7 units from S and 7 units from
T, find the co-ordinates of S and T.
hence there are 2 points on the x axis that are 7
units from P
i.e. S and T
and
45
The position vectors of the points P and Q are
p i 3j 4k and q 7i j 5k
respectively. a) Express PQ in component
form. b) Find the length of PQ.
46
PQR is an equilateral triangle of side 2 units.
Evaluate a.(b c) and hence identify two
vectors which are perpendicular.
120o
NB for a.c vectors must diverge ( so angle is
120 )
so, a is perpendicular to b c
Hence
47
Calculate the length of the vector 2i 3j ?3k
Length
48
Find the value of k for which the vectors
and are perpendicular
Put Scalar product 0

49
A is the point (2, 1, 4), B is (7, 1, 3) and C
is (6, 4, 2). If ABCD is a parallelogram, find
the co-ordinates of D.
The journey B to A is the same as the journey
from C to D
That journey from C gives D(6 -5, 4 -2,
2 1)
D(11, 2 , 3)
50
The vectors a, b and c are defined as
follows a 2i k, b i 2j k,
c j k a) Evaluate a.b a.c b) From
your answer to (a), make a deduction about the
vector b c
a)
b)
b c is perpendicular to a
51
In the square based pyramid, all the eight edges
are of length 3 units.
Evaluate p.(q r)
Triangular faces are all equilateral
52
A and B are the points (-1, -3, 2) and (2, -1,
1) respectively. B and C are the points of
trisection of AD. That is, AB BC CD. Find the
coordinates of D
(-1,-3,2)
(2,-1,1)
? D is (-1 9, -3 6, 2 3)
? D is (8, 3, 1)
53
The point Q divides the line joining P(1, 1, 0)
to R(5, 2 3) in the ratio 21. Find the
co-ordinates of Q.
(5,2,-3)
(-1,-1,0)
? Q is (-1 4, -1 2, 0 2)
? Q is (3, 1, 2)
54
VABCD is a pyramid with rectangular base ABCD. VA
7i 13j 11k, AB 6i 6j 6k, AD
8i 4j 4k K divides BC in the
ratio 13. Find VK in component form.
¼BC
AB
VA
VK
¼AD
AB
VA
7i 13j 11k
6i 6j 6k
2i j k
i 8j 18k
55
A is the point (2, 5, 6), B is (6, 3, 4) and
C is (12, 0, 1). Show that A, B and C are
collinear and determine the ratio in which B
divides AC
AB and BC are scalar multiples, so are parallel.
B is common. A, B, C are collinear
B divides AB in ratio 2 3