3 18.1 vectors and vector notation - mrvahora · 3 18.1 vectors and vector notation 355a 1 on...
TRANSCRIPT
�3 18.1 Vectors and vector notation
355A
1 On Resource sheet 18.1a, draw accurately and label the following vectors.
a Vector AB� ��
with magnitude 5 cm and direction due north.
b Vector MN� ���
with magnitude 4 cm and direction south west.
c Vector a magnitude 4 cm and direction due west.
d Vector b magnitude 8 cm and direction 060°.
e Vector c magnitude 5.5 cm and direction 145°.
f Vector d magnitude 5.5 cm and direction 230°.
2 Write each vector as a column vector.
PQ� ���
= DC� ���
= EF� ���
= GH� ���
=
...................................... ...................................... ...................................... ......................................
a = b = c =
...................................... ...................................... .....................................
.
d = e = f =
...................................... ...................................... ......................................
3 On Resource sheet 18.1b, draw the following vectors. Label each vector.
a AB� ��
= 23
b CD� ���
= 41−
c EF� ���
= −
−
33
d a = 02
e b = 40
f c = 11−
g d = −
30
h MN� ���
= 04−
Guided practice worksheet
–4Remember: ‘magnitude’ means size. The vector ( ) means move 4 left and 2 up. 2
3A
B= ( ) because from A to B is 3 right, 2 down. –2
A
BP
Q
C
D
E FG
H
f
a bc
d
e
A
Questions are targeted at the grades indicated
�3 18.1 Vectors and vector notation
355B
Guided practice worksheet
4 Using Resource sheet 18.1b, for each pair of points below:i plot the points on the axes ii write down a vector joining the two points.
a A(4, 1), B(8, 2) b C(0, 1), D(1, 0) c E(1, 4), F(1, 8) d G(0, 3), H(3, 1)
e I(3, 3), J(4, 8) f K(5, 8), L(8, 8) g M(8, 3), N(5, 6)
5 a Plot the following points on Resource sheet 18.1c.
A(1, 2), B(2, 6), C(5, 6), D(5, 0)
b Draw the vectors AB� ��
, BC� ���
, DC� ���
and AD� ���
c Write each vector as a column vector.
6 a Plot these points on the same grid in Resource sheet 18.1c.
E(3, 2), F(2, 6), G(5, 4), H(6, 0)
b Write down the vectors
i EF� ���
ii HG� ���
iii EH� ���
iv FG� ���
c What do you notice about these vectors?
d What kind of shape is EFGH?
7 PQRS is a square. P is the point (5, 4). PQ� ���
= −
32
and PR� ��
= −
−
23
a Plot P, Q and R on Resource sheet 18.1c.
b Mark the vectors PQ� ���
and PR� ��
.
c Mark the point S so that PQRS is a square.
d i Write down the vectors SR� ��
and PQ� ���
.
ii What do you notice about these vectors?
8 These are the end points of three equal vectors.
a Join up the pairs of points with equal vectors.
b Write down the column vectors of these pairs of points. ........................................................................................................
y
x1 2 3 4 5 60
1
2
3
4
5
6
A
1 Calculate the magnitude of each vector i as a surd ii correct to 1 decimal place, where necessary.
Use the correct notation to write your answer, e.g. | a | = , | AB� ��
| =, or AB =
a a = 14
i ..................................................................
ii ..................................................................
b AB� ��
= 02
i ..................................................................
ii ..................................................................
c PQ� ���
= 512
i ..................................................................
ii ..................................................................
d d = −
23
i ..................................................................
ii ..................................................................
e e = 41−
i ..................................................................
ii ..................................................................
f MN� ���
= 34−
i ..................................................................
ii ..................................................................
g f = −
60
i ..................................................................
ii ..................................................................
h DE� ���
= 11
i ..................................................................
ii ..................................................................
2 Using Resource sheet 18.2, for each pair of points:
i plot the points on the axes
ii write down a vector joining the two points
iii find the magnitude of the vector, correct to 1 decimal place.
a A(5, 2), B(7, 8) b C(5, 7), D(4, 6) c E(0, 7), F(3, 8) d G(0, 6), H(6, 0)
e I(7, 2), J(7, 0) f K(4, 1), L(0, 1) g M(1, 2), N(1, 4)
�3 18.2 The magnitude of a vector
357
Guided practice worksheet
xRemember: the magnitude (length) of the vector ( ) is √ x2 + y2
y
–5J K = ( ) 3
(i) | J K | = √ (–5)2 + 32 = √ 25 + 9 = √ 34 (we could just write JK = √ 34)
(ii) | J K | = 5.8 (1 d.p.)
A
�3 18.3 Addition of vectors
359A
1 The diagram shows some vectors.
Draw the following vector sums on the squared paper below. Write each vector sum as a column vector.
a a + b ................................................ b b + d ................................................
c a + c ................................................ d e + d ................................................
e c + e ................................................ f b + b ................................................
g a + e + d ........................................ h a + b + c ........................................
Guided practice worksheet
a + dJoin a to d so that their arrows point in the same direction. 2a + d = ( ) 1
a
b
cd
e
ad
a + d
A
Questions are targeted at the grades indicated
�3 18.3 Addition of vectors
359B
Guided practice worksheet
2 Work out the following vector sums.
a 25
13
+
.........................................................
b 03
30
+
.........................................................
c −
14
52
+
.........................................................
d 42
32−
−
+
.........................................................
e −
−
−
−
31
53
+
.........................................................
f 33
55
−
−+
.........................................................
g 24
14
−
−+
.........................................................
h −
−
11
11
+
.........................................................
3 Given that a = 23
, b = −
14
and c = 02−
, � nd as a column vector
a a + b b b + c
......................................................... .........................................................
c a + c d a + a
......................................................... .........................................................
e a + b + c f b + b + c
......................................................... .........................................................
g a + c + c
.........................................................
4 For each pair of vectors AB� ��
and BC� ���
:
i make a rough sketch of the vectors
ii draw vector AC� ���
iii write AC� ���
as a column vector.
a AB� ��
= 22
and BC
� ��� = 1
0
.........................................................
1 –3A
B = ( ) and B
C ( ) –2 3
1 –3 –2A
C = ( ) + ( ) = ( ) –2 3 1
C
A
B
A
�3 18.3 Addition of vectors
359C
Guided practice worksheet
b AB� ��
= 02
and BC� ���
= 30
.........................................................
c AB� ��
= −
30
and BC� ���
= 03
.........................................................
d AB� ��
= 43
and BC� ���
= −
10
.........................................................
e AB� ��
= −
−
22 and BC
� ��� = −
34
.........................................................
f AB� ��
= 04−
and BC� ���
= 42
.........................................................
A
�3 18.4 Parallel vectors
361A
1 The diagram shows some vectors.
Draw the following vector sums on the squared paper below. Write each vector sum as a column vector.
a 3a .................................................................. b 2b ..................................................................
c –a .................................................................. d 12e ..................................................................
e –2a .................................................................. f 4d ..................................................................
g –3b .................................................................. h − 12 c ..................................................................
Guided practice worksheet
a bc d e
2a has twice the magnitude of a and is in the same direction.
–b has the same magnitude of b but is in the opposite direction.
4 02a = ( ) –b = ( ) 2 1
a
a2a
2a = (42) –b = (01)
–b
A
Questions are targeted at the grades indicated
�3 18.4 Parallel vectors
361B
Guided practice worksheet
2 Given that a = 13
, b = 46−
and c = −
20
, � nd the following as a column vector.
a 4a .................................................................. b –b ..................................................................
c –3a .................................................................. d 4c ..................................................................
e 12b .................................................................. f − 1
2 c ..................................................................
g 2a + c .................................................................. h a + 3c ..................................................................
i 2b + 2c .................................................................. j a – c ..................................................................
k b – c .................................................................. l 2a – c ..................................................................
m b – 2a .................................................................. n –a – c ..................................................................
o –2b + 3c ..................................................................
3 Find the pairs of parallel vectors.
21
03
93
22
44
55
− − −
11
42
21
42
09
31
− −
........................................................................................................................................
........................................................................................................................................
........................................................................................................................................
........................................................................................................................................
4 –2 x 4 –8–2b = –2 x ( ) = ( ) = ( ) –6 –2 x –6 12
–3 –6 –6 –3( ) is parallel to ( ) because ( ) = 2 x ( ) 2 4 4 2
A
�3 18.4 Parallel vectors
361C
Guided practice worksheet
4
a Draw the position vector of each point on the diagram.
b Write down the position vectors as column vectors.
.................................................................. ..................................................................
.................................................................. ..................................................................
c Complete the following:
AB� ��
= AO� ���
+ OB� ��
= ....................................... + ....................................... = .......................................
CD� ���
= CO� ���
+ ....................................... = ....................................... + ....................................... = .......................................
d What can you say about the lines AB and CD?
..................................................................
..................................................................
y
x1 2 3 4 5 60
1
2
3
4
5
6
A
B
DC
Hint join the origin O to the point.
A*
�3 18.5 Solving geometric problems in two dimensions
363A
1 From the given information, decide whether points A, B and C lie on a straight line.
a AB BC� �� � ���
=
=
102
51
,
..............................................................................
b AB BC� �� � ���
=
=
126
32
,
..............................................................................
c AB BC� �� � ���
=
=
03
60
,
..............................................................................
d AB BC� �� � ���
=
=
−
−33
66
,
..............................................................................
e AB BC� �� � ���
=
=
− −
12
32
12,
..............................................................................
f AB BC� �� � ���
=
=
−20
50
,
..............................................................................
g AB BC� �� � ���
=
=
−164
41
,
..............................................................................
h AB BC� �� � ���
=
=
−
−43
43
,
..............................................................................
2 From the given information, decide whether points P, Q and R lie on a straight line.
a PQ� ���
= a + b, PR
� ��
= 6a + 6b ............................................................................................................................................................................
.................................................................................................................................................................................................................................
b PQ� ���
= a – 3b, PR
� ��
= 2a – 6b .........................................................................................................................................................................
.................................................................................................................................................................................................................................
c PQ� ���
= 3a + 2b, PR
� ��
= 9a + 5b .......................................................................................................................................................................
.................................................................................................................................................................................................................................
Guided practice worksheet
6 2A
B = ( ) B
C = ( ) –3 –1
A, B and C will lie on a straight line if A
B and B
C have the same direction.
6 2 ( ) = 3 × ( ) and so A
B and B
C are parallel. Also, they meet at B. –3 –1
A, B and C lie on a straight line.
P
Q = 1–2 a + b, P
R = 2a + 6b
P, Q and R will lie on a straight line if P
Q and P
R have the same direction.
4 × P
Q = 4 × ( 1–2 a + b) = 2a + 4b which is not parallel to P
R = 2a + 6b.
P, Q and R do not lie on a straight line.
A*
Questions are targeted at the grades indicated
�3 18.5 Solving geometric problems in two dimensions
363B
Guided practice worksheet
d PQ� ���
= 3a – 2b, PR
� ��
= 6a + 9b ......................................................................................................................................................................
.................................................................................................................................................................................................................................
e PQ� ���
= 4a + 6b, PR
� ��
= 6a + 9b .......................................................................................................................................................................
.................................................................................................................................................................................................................................
f PQ� ���
= – a + 2b, PR
� ��
= 2a – 4b .....................................................................................................................................................................
.................................................................................................................................................................................................................................
g PQ� ���
= 4a – b, PR
� ��
= –2a + 1
2 b ......................................................................................................................................................................
.................................................................................................................................................................................................................................
h PQ� ���
= 5a + 2b, PR
� ��
= a + b ............................................................................................................................................................................
.................................................................................................................................................................................................................................
3 The diagram shows the position vectors of points A, B and C from O.
a Find i AB� ��
...........................................................................................
ii BC� ���
...........................................................................................
b Show that A, B and C lie on the same straight line.
.........................................................................................................
.........................................................................................................
4 In the vector diagram, DC� ���
= 1
3 AB� ��
.
a Find
i DC� ���
...........................................................................................
ii AD� ���
...........................................................................................
Line BC is extended to point E so that CE = 12 BC.
b Find
i CE� ��
...........................................................................................
ii DE� ���
...........................................................................................
c What can you say about points A, D and E? Give reasons for your answers.
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
A
O
B
Ca
a + 2b
2a + 3b
A D E
C
B
12a
6b
A*