T Madas Translation Sliding vector Horizontal Steps Vertical

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Translation = Sliding vector Horizontal Steps Vertical Steps I O © T Madas

Translation = Sliding vector Horizontal Steps Vertical Steps I A vector: is a line with a start and a finish. A vector has a direction and a length. O © T Madas

Translation = Sliding vector Horizontal Steps Component Vertical Steps Component A vector: is a line with a start and a finish. A vector has a direction and a length. © T Madas

Translation = Sliding vector Horizontal Steps Component Vertical Steps 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. 2. 3. line of action a direction a given size (magnitude) A B © T Madas

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 bold or underlined when handwritten In component form, if the vector is drawn on a grid: 4 5 © T Madas

F E B D H A C G © T Madas

B D AB = 4 5 A -5 CD = 4 C © T Madas

4 0 4 = + 5 0 5 B AB = 4 5 0 5 A 4 0 © T Madas

-5 0 = -5 + 0 4 4 D 0 4 CD = -5 4 C -5 0 © 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 ? C 8 AC = 6 3 BC = 4 B AB + BC = AC A 5 AB = 2 5 3 8 = + 2 4 6 © T Madas

To add vectors when written in component form: we add the horizontal components and the vertical components of the vectors separately. C 8 AC = 6 3 BC = 4 B AB + BC = AC A 5 AB = 2 5 3 8 = + 2 4 6 © T Madas

3 Let the vector u = What is the vector 2 u ? 2 2 xu 2 x 6 3 2 x 3 = = 4 2 2 x 2 3 u= 2 To multiply a vector in component form by a number (scalar), we multiply each vector component by that number. 2 u = 6 4 © T Madas

© T Madas

7 -4 5 Let the vectors u = , v= and w = 1 2 3 Calculate: 5 7 12 1. u + v + = 3 1 4 v u v + u © T Madas

7 -4 5 Let the vectors u = , v= and w = 1 2 3 Calculate: 5 7 -4 8 2. u + v + w + + = 3 1 2 6 + uu v + w w v © T Madas

7 -4 5 Let the vectors u = , v= and w = 1 2 3 Calculate: 5 -4 5 -8 -3 3. u + 2 w +2 + = = 3 2 3 4 7 2 w u+ 2 w u © T Madas

7 -4 5 Let the vectors u = , v= and w = 1 2 3 Calculate: 5 7 5 -7 -2 4. u – v – + = = 3 1 3 -1 2 u -v – v v u © T Madas

7 -4 5 Let the vectors u = , v= and w = 1 2 3 Calculate: 5 -4 5 4 9 5. u – w – + = = 3 2 3 -2 1 w -w u u–w © T Madas

© T Madas

x What is the magnitude of vector r = ? y x r= y d 2 = x 2 + y 2 c r = x 2+y 2 d = x 2 + y 2 x © T Madas

x What is the magnitude of vector r = ? y r = x 2+y 2 8 a= 6 a = 82 + 62 = 64 +36 = 100 = 10 units -5 b= 12 2 ( ) b = -5 +122 = 25 +144 = 169 = 13 units -6 u= -2 2 ( ) u = -6 +(-2)2 = 36 + 4 = 40 ≈ 6. 3 units 7 v= 24 v = 72 + 242 = 49 + 576 = 625 = 25 units © T Madas

© T Madas

7 An object is translated using the vector followed by 3 -5 a second translation by the vector. 4 Work out the vector for the combined translation. 7 -5 2 + = 3 4 7 2 7 -5 4 7 3 © T Madas

© T Madas

An object is placed at the origin of a standard set of axes and is subject to four successive translations using the following vectors: 7 , -6 , -1 and 5. 8 3 -6 7 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. 7 -6 -1 5 5 + + + = 8 3 -6 7 12 magnitude = 52 + 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. y The point C is such so that: • AC is parallel to • AB = BC 0 -1 A -2 2. Write AC as a column vector. AB = -3 B x The point D is such so that: O • ABCD is a rhombus 3. Calculate the area of the rhombus. -2 AB = -3 AB = (-2)2 + (-3)2 = 4+ 9 = 13 ≈ 3. 6 units © 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. y The point C is such so that: • AC is parallel to • AB = BC A 0 -1 2. Write AC as a column vector. The point D is such so that: • ABCD is a rhombus 0 -1 B 0 AC = -6 O x C 3. Calculate the area of the rhombus. © 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. y The point C is such so that: • AC is parallel to • AB = BC 0 -1 2. Write AC as a column vector. The point D is such so that: • ABCD is a rhombus A 3 3 B D 12 3 3 O x C 3. Calculate the area of the rhombus. © T Madas

© T Madas