the Further Mathematics network
www.fmnetwork.org.uk
1
the Further Mathematics network
www.fmnetwork.org.uk
Further Pure 3: Teaching
Vector Geometry
Let Maths take you Further…
2
Overview
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Scalar and vector product
Basic skill sets related to lines and planes
Applications of scalar and vector product
The scalar triple product, determinants,
equations and matrices
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Scalar Product
⎛ 2⎞ ⎛ 3⎞
⎜ ⎟⎜ ⎟
⎜ 3 ⎟ . ⎜ 2 ⎟ = (2 × 3) + (3 × 2) + (4 × 1) = 16
⎜ 4⎟ ⎜1⎟
⎝ ⎠⎝ ⎠
Get students to get a feel for the scalar
product
Work in two dimensions to begin with –
when is it negative, when is it positive,
when is it zero?
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Use of scalar product in lighting
Angle between
normal vector
and light
source
determines
how light the
surface should
appear
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Use of scalar product in lighting
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Use of scalar product in lighting
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Use of scalar product in lighting
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Use of scalar product in lighting
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Lighting a Plane
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Vector Product
⎛ 2 ⎞ ⎛ 3 ⎞ ⎛ (3 × 1) − (2 × 4) ⎞ ⎛ −5 ⎞
⎜ ⎟ ⎜ ⎟ ⎜
⎟ ⎜ ⎟
3
×
2
=
−
(2
×
1)
−
(3
×
4)
[
]
⎜ ⎟ ⎜ ⎟ ⎜
⎟ = ⎜ 10 ⎟
⎜ 4 ⎟ ⎜ 1 ⎟ ⎜ (2 × 2) − (3 × 3) ⎟ ⎜ −5 ⎟
⎝ ⎠ ⎝ ⎠ ⎝
⎠ ⎝ ⎠
Also |a × b| = |a||b|sinθ
And a × b is perpendicular to both of a
and b in accordance with the right hand
rule.
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Vector Product
„
The fact that |a × b| = |a||b|sinθ means that
the length of a × b is the same as the area of
the parallelogram made using a and b as the
sides.
a
b
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Use of vector product in video
games, what’s behind what?
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Use of vector product
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The scalar and vector product
„
a.b = | a | | b | cosθ
a
θ
|a.b|/|a|
|a.b|/|b|b
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Or if | b |= 1
„
a.b = | a | | b | cosθ
a
θ | a.b |
b
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The scalar and vector product
„
| a x b | = | a | | b | sinθ
a
| a x b |/ |b|
| a x b | /|a|
θ
b
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Or if | b |= 1
„
| a x b | = | a | | b | sinθ
|a x b |
a
θ
b
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In summary
To resolve a parallel and perpendicular to b
If b is the unit vector of b then
„
„
a resolved parallel to b is | a.b |
a resolved perpendicular to b is | a x b |
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Quick Skills Test
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Resolve ⎛ 2 ⎞ in the direction of ⎛ 1 ⎞
„
Resolve ⎛ 2 ⎞ perpendicular to ⎛ 1 ⎞
⎜ ⎟
⎜ 3⎟
⎜5⎟
⎝ ⎠
⎜ ⎟
⎜ 3⎟
⎜5⎟
⎝ ⎠
⎜ ⎟
⎜1⎟
⎜ 2⎟
⎝ ⎠
⎜ ⎟
⎜1⎟
⎜ 2⎟
⎝ ⎠
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The basics
„
„
The vector equation of
a line
Conversion between
Cartesian and vector
form of line
⎛1⎞
⎛ 2⎞
r =⎜ ⎟+λ⎜ ⎟
⎝ 2⎠
⎝ 3⎠
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Examples of short questions to
ask students
„
„
„
⎛1⎞
⎛ 2⎞
Is (3, 2) on the line r = ⎜ ⎟ + λ ⎜ ⎟
⎝ 2⎠
⎝ 3⎠
⎛ 2⎞
⎛ −4 ⎞
?
⎛1⎞
⎛ 2⎞
Are the lines s = ⎜ ⎟ + μ ⎜ ⎟ , r = ⎜ ⎟ + λ ⎜ ⎟
⎝ 2⎠
⎝ 3⎠
⎝ 2⎠
⎝ −6 ⎠
parallel?
Find the line which has the points (1, 4) and
(7, 10) on it.
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The basics
„
The vector equation of
a line
⎛1⎞
⎛2⎞
⎜ ⎟
⎜ ⎟
r = ⎜ 2⎟ + λ ⎜ 3 ⎟
⎜ 4⎟
⎜ −1 ⎟
⎝ ⎠
⎝ ⎠
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Introducing Planes - Ideas
„
If the Cartesian equation of a line is given in the
form ax + by = d
then
⎛a⎞
⎜ ⎟
⎝b⎠
is perpendicular to the line and as d changes
the height of the line changes.
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Planes
⎛ 3⎞
⎜ ⎟
⎜ 2⎟
⎜ 4⎟
⎝ ⎠
⎛3⎞
⎜ ⎟
⎜1⎟
⎜2⎟
⎝ ⎠
( 1, 1, 2 )
( x, y , z )
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Basics – list
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Vector and Cartesian form of equations and
planes and how to convert between them.
Finding a line given two points on it
Finding a plane given three points on it
Finding the intersection of two lines.
Finding the intersection of a line and a plane
Finding the intersection of two planes
Finding the intersection of three lines and
geometrical interpretation
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Testing the basics
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Basics jigsaw
Timed tests
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Applications of the scalar
and vector product
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Distance between a point and a line
Distance between two skew lines
Distance of a point from a plane
Scalar Triple Product and applications
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Distance between a point and
a line
„
We can see this is just AP resolved
perpendicular to the direction of l.
l
P
A
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Distance between skew lines
„
We can see that this is just the vector
between any point on l1 and any point on l2
resolved in the direction perpendicular to both
lines
P
A
l1
l2
Q
B
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Distance of a point from a
plane
„
This is just RP resolved parallel to the normal
vector where R is any point on the plane.
P
normal
R
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The scalar triple product
„
„
„
„
Where a, b and c are vectors this means
a.(b x c).
We know that a.(b x c) = |a||b x c|cosθ where
θ is the angle between a and b x c.
|b x c| is the area of the base of the
parallelopiped and the modulus of |a|cos θ is
its vertical height.
Therefore |a.(b x c)| is the volume of the
parallelopiped formed from a, b and c
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Applications
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Testing whether two lines meet
Testing whether four points are coplanar
Proving that the determinant gives the scale
factor of volume enlargement for
transformations of three dimensional space.
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Collision Detection
„
Asteroids was one of the first video games
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Using vectors in
Collision Detection – 2D
C
β
α
When P is inside the
triangle ABC,
P
α + β + γ = 360°
γ
P
B
C
A
When P is
outside the
triangle ABC
α
β
γ
α + β + γ < 360°
A
B
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Geogebra Demo
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Video Games Vector Tricks
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Exam Question
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Examiners’ Comments
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Exam Question
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Examiners’ Comments
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