FP3 Chapter 5 Vectors
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Overview of FP3 Vectors
B: Area of shapes
Triangle
Parallelogram
E: Eqns of planes
F: Applications
Intersection of lines/planes
Angle between lines/planes
Shortest distances
Volume of tetrahedron
Volume of parallelepiped
Disclaimer: I actually like vectors.
Vector/Cross Product
Recall from C4 that the dot product between two vectors is as follows:
(The dot product is clearly useful for finding the angle between two vectors)
Vector/Cross Product
Inputs
Output
Angle between them
(with some manipulation)
Inputs
Output
Vector perpendicular to them
(with some magnitude)
This why they’re known as the scalar and vector products, because we get a scalar and a vector as an output respectively.
Examples
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Determining Cross Product
Determining Cross Product using Determinants
(note the minus for the middle one)
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Determining Cross Product using Determinants
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Examples
? vector directly
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Recall that if two vectors are perpendicular, their dot product is 0.
Further Example
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Test Your Understanding
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Yes, your Silver Casios calculate cross products…
This is the answer we’re expecting…
Exercise 5A
One last thing…
We can technically use the cross product to find the angle between two lines (although we’d usually use the dot product):
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B: Area of shapes
Triangle
Parallelogram
E: Eqns of planes
F: Applications
Intersection of lines/planes
Angle between lines/planes
Shortest distances
DESCENDING
Volume of tetrahedron
Volume of parallelepiped
Area of a triangle
OMG
it’s double the area of a triangle.
Area of a triangle
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Bro Note: I see no reason to use this expanded form. Just use the second line!
Examples
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Test Your Understanding
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Exercise 5B
B: Area of shapes
Triangle
Parallelogram
E: Eqns of planes
F: Applications
Intersection of lines/planes
Angle between lines/planes
Shortest distances
DESCENDING
Volume of tetrahedron
Volume of parallelepiped
Triple Scalar Product
(Recall that ‘scalar product’ is another name for dot product)
The Triple Scalar Product of three vectors is the dot product of one of the vectors with the cross product of the other two.
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? which could be expressed more simply as…
Examples
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Volume of Parallelepiped
(pronounced “parallel-uh-pie-pidd”. You’re welcome)
perp to base
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Volume of Tetrahedron
Similarly, the formula for a volume of a pyramid still works if the pyramid is not ‘right’, i.e. the apex (peak) is not above the centre of the base.
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Test Your Understanding
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(We saw (a) and (c) of this question earlier)
Exercise 5C
B: Area of shapes
Triangle
Parallelogram
Volume of tetrahedron
E: Eqns of planes
F: Applications
Intersection of lines/planes
Angle between lines/planes
Shortest distances
DESCENDING
Volume of parallelepiped
Vector equations of lines
This chapter is effectively divided into two.
The first, which we’ve now completed, concerns the application of cross products and triple scalar products to finding areas and volumes.
This second half of the chapter concerns the application of cross products (and dot products) to equations of lines and planes. These tend to be more common in exams.
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Vector equations of lines
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In Edexcel formula booklet:
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Test Your Understanding
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Exercise 5D
B: Area of shapes
Triangle
Parallelogram
Volume of tetrahedron
E: Eqns of planes
F: Applications
Intersection of lines/planes
Angle between lines/planes
Shortest distances
DESCENDING
Volume of parallelepiped
Planes
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Example
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Plane using vectors/lines on plane
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Test Your Understanding
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Vector form of equation of plane
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Examples
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Test Your Understanding
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Exercise 5E
B: Area of shapes
Triangle
Parallelogram
Volume of tetrahedron
E: Eqns of planes
F: Applications
Intersection of lines/planes
Angle between lines/planes
Shortest distances
DESCENDING
Volume of parallelepiped
Section Overview
Intersection of:
Angle between:
(Shortest) Distance between:
Points of intersection of lines/planes
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The essence of this skill is just solving simultaneously, because this gives us points satisfying both equations and hence are on both.
Test Your Understanding
We did this last lesson.
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Points of intersection of lines/planes
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Alternative (and much much nicer) Method
This method is the ‘primary’ method in mark schemes and is considerably more elegant/less clunky:
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Test Your Understanding
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Exercise 5F
Q1, 2, 3
(you won’t be able to do the rest yet)
Section Overview
Intersection of:
Angle between:
(Shortest) Distance between:
Angle between two lines (C4 recap)
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Angle between line and plane
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Test Your Understanding
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Angle between two planes
CATCHPHRASE INTERMISSION�“Say what you see!”
Snakes on a Plane
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Test Your Understanding
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Exercise 5F
Q4-8
(you won’t be able to do Q9 onwards yet)
Section Overview
Intersection of:
Angle between:
(Shortest) Distance between:
Shortest Distance from Plane to Origin
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Shortest Distance from Plane to Origin
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Shortest Distance from Plane to Origin
b)
Shortest Distance from Plane to Origin
c)
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Test Your Understanding
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Length of Line after a ‘Projection’
Curiously this is never explained in the textbook but the theory is used in multiple subsequent examples…
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Memory Tip: You divide by the magnitude of the vector you are projecting onto.
Shortest Distance between Line and Line
Method 1: Using calculus
This avoids having to differentiate an expression with a square root in it.
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Shortest Distance between Line and Line
Method 2: Using a projection
Use the two known points on the lines and form a triangle:
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Shortest Distance between Skew Lines
Bro Definition: Skew Lines are lines which are not parallel and do not cross.
General point on line
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Example
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Test Your Understanding
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Shortest Distance between Line and Point
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Bro Tip: Probably the fastest/easiest method.
Exercise 5F
Q9 onwards
B: Area of shapes
Triangle
Parallelogram
E: Eqns of planes
F: Applications
Intersection of lines/planes
Angle between lines/planes
Shortest distances
DESCENT COMPLETED
Volume of tetrahedron
Volume of parallelepiped