© t madas. translation sliding vector horizontal steps vertical steps = o i
TRANSCRIPT
© T Madas
© T Madas
Translation Sliding5
3
æöç ÷ç ÷ç ÷ç ÷è øvector
Horizontal Steps
Vertical Steps
A B
CD
=
A¢ B¢
C¢D¢
O
I
© T Madas
Translation Sliding5
3
æöç ÷ç ÷ç ÷ç ÷è øvector
Horizontal Steps
Vertical Steps
A B
CD
=
A¢ B¢
C¢D¢
O
IA vector:is a line with a start and a finish.
A vector has a direction and a length.
© T Madas
Translation Sliding5
3
æöç ÷ç ÷ç ÷ç ÷è øvector
Horizontal Steps
Vertical Steps
=
Component
Component
A vector:is a line with a start and a finish.
A vector has a direction and a length.
© T Madas
Translation Sliding5
3
æöç ÷ç ÷ç ÷ç ÷è øvector
Horizontal Steps
Vertical Steps
=
Component
Component
If a vector is drawn on a grid we can always write it, in component form.
© T Madas
© T Madas
A vector is a line with a start and a finish.It therefore has:
1. line of action2. a direction3. a given size (magnitude)
A B ABuuur
= a
A B BAuur
-= a
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a
ABuuur
a
we write vectors in the following ways:By writing the starting point and the finishing point in capitals with an arrow over them
With a lower case letter which:is printed in boldor underlined when handwrittenIn component form, if the vector is drawn on a grid:45
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E
F
A
B
C
D
G
H
Let AB = auuur
CD =uuur
2a
EF =uur
12a
HG =uuur
-2a
© T Madas
45
A
B
-5 4
C
D
AB =CD =
© T Madas
45
40
05
40
05
+ = 45
A
B
AB =
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-5 4
-5 0
04
-5 0
04
+ = -5 4
C
D
CD =
© T Madas
What is the vector from A to B ?What is the vector from B to C ?What is the vector from A to C ?
A
B
C
52
34
86
52
34
+ = 86
AB =
BC =AC =
AB + BC = AC
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A
B
C
52
34
86
52
34
+ = 86
AB =
BC =AC =
AB + BC = AC
To add vectors when written in component form:we add the horizontal components and the vertical components of the vectors separately.
© T Madas
32
Let the vector u =
32
32
2 x =
u =
u2 x
What is the vector 2u ?
64
2u =
2 x 32 x 2 =
64
To multiply a vector in component form by a number (scalar), we multiply each vector component by that number.
© T Madas
© T Madas
+
Let the vectors u = , v = and w = 53
71
-4 2
Calculate:
1. u + v
u
v
53 =
71
124
u + v
© T Madas
+
Let the vectors u = , v = and w = 53
71
-4 2
Calculate:
2. u + v + w
u
v
53 =
71
86
u + v
+ w
+-4 2
w
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+ 2
Let the vectors u = , v = and w = 53
71
-4 2
Calculate:
3. u + 2w
u
53 =
-3 7
u +
2w
-4 2
2w
+53
-8 4=
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–
Let the vectors u = , v = and w = 53
71
-4 2
Calculate:
4. u – v
u
53 =
-2 2
u – v
71
-v
+53
-7-1=
v
© T Madas
–
Let the vectors u = , v = and w = 53
71
-4 2
Calculate:
5. u – w
u
53 =
91
u – w
-4 2
-w
+53
4-2=
w
© T Madas
© T Madas
What is the magnitude of vector r = ?xy
x
yd
xy
r =d
2 = x 2 + y
2
d = x 2 + y
2
r = x 2+y
2
© T Madas
( )2-5-512
What is the magnitude of vector r = ?xy
r = x 2+y
2
a =86 a = 82 +62 = 64+36 = 100= 10 units
b = b = +122= 25+144= 169= 13 units
( )2-6-6-2
u = u = +(-2)2 = 36+4 = 40≈ 6.3 units
v = 724
v = 72 +242 = 49+576= 625= 25 units
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© T Madas
An object is translated using the vector
followed by a second translation by the vector
.
Work out the vector for the combined
translation.
73
-5 4
+73 =
-5 4
27
73
-5 42
7
© T Madas
© T Madas
52
=512+
57+
-1-6
An object is placed at the origin of a standard set of axes and is subject to four successive translations using the following vectors: , , and .
1. Work out the single vector that could be used to produce the same result as these four translations.2. Calculate the magnitude of this vector.
78
-6 3
-1-6
57
+78
-6 3
magnitude= +122 = 25+144= 169= 13 units
© T Madas
© T Madas
The points O (0,0), A (1,5) and B (-1,2) are given.
1. Write AB as a column vector and calculate its magnitude.The point C is such so that:
• AC is parallel to
• AB = BC
2. Write AC as a column vector.
The point D is such so that:
• ABCD is a rhombus
3. Calculate the area of the rhombus.
0-1
x
y
O
A
B
-2-3AB =
( )2-2-2-3AB = AB = + (-3)2 = 4+ 9 = 13≈ 3.6 units
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The point C is such so that:
• AC is parallel to
• AB = BC
2. Write AC as a column vector.
The point D is such so that:
• ABCD is a rhombus
3. Calculate the area of the rhombus.
0-1
The points O (0,0), A (1,5) and B (-1,2) are given.
1. Write AB as a column vector and calculate its magnitude.
x
y
O
A
B
0 -1
C
0-6AC =
© T Madas
The point C is such so that:
• AC is parallel to
• AB = BC
2. Write AC as a column vector.
The point D is such so that:
• ABCD is a rhombus
3. Calculate the area of the rhombus.
0-1
The points O (0,0), A (1,5) and B (-1,2) are given.
1. Write AB as a column vector and calculate its magnitude.
x
y
O
A
B
C
D
3 3
33
12
© T Madas