c4: chapter 5 – vectors dr j frost ([email protected]) last modified: 11 th february...
TRANSCRIPT
RECAP: GCSE stuff
A vector has both direction and magnitude. It represents a movement.
(𝑥𝑦 ) (𝑥 , 𝑦 )
𝐴 𝐵�⃗�𝐵
Vector Coordinate
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Vectors that are equal have both the same magnitude and direction.
𝑂
𝐴
𝐵
𝑀
�⃗�𝑀=12𝒂+𝒃
𝒂
𝟐𝒃
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Basics #1: Magnitude of a Vector
! The magnitude of a vector is its length.
(34)
𝐴
𝐵
|⃗𝐴𝐵|=√32+42=53
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|( 1−1)|=√2 |( −5−12)|=13? ?
𝒂=(341)|𝒂|=√32+42+12=√26
𝒃=(−203 )|𝒃|=√22+02+32=√13
If If ?
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Basics #2: Scalars
! A scalar is a quantity which scales a vector (hence the name). It is just an ordinary number.
2𝒂𝒂scalar
We typically use (“mu”) and (“lambda”) as constant letters for scalars e.g.
Scalars are also used in the context of directionless quantities, e.g. velocity is a vector while speed is a scalar.
Basics #3: Unit Vectors
! A unit vector is a vector whose magnitude is 1
There’s certain operations on vectors that require the vectors to be ‘unit’ vectors. We just scale the vector so that its magnitude is now 1.
(34)→ 15 (34) (11)→ 1
√2 (11)
( 10−1)→ 1√2 ( 10−1) 𝒂→
𝒂|𝒂|
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Basics #4: Parallel Vectors
! If is a vector, then represents a vector parallel to , where is a scalar.
Example: Show that and are parallel.
𝒃
𝒂
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Basics #5: Uniqueness of composition
! If , and the non-zero vectors and are not parallel, then and .
𝒂
𝒃
Suppose we want to find in terms of and .1. Using just some linear combination of and (i.e. ) is it always possible to
reach ? Yes, provided and are not parallel and non-zero.2. Are and unique, in the sense there is only one combination of amounts
of and we can use to get to ? Yes, they are unique.(This is obvious for example if and point in the directions of the and axis. A coordinate is uniquely identified by its and value)
𝑶
𝑷
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Basics #5: Uniqueness of composition
Given that , where and are non-zero, non-parallel vectors, find the values of the scalars and .
2𝑠+𝑡=5?
Basics #6: Position VectorsIf a point has coordinate , what vector would we use to get there from the origin?
𝐴 (4,3 )𝑎=(43)
𝑂
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! The position vector of a point is the vector, where is the origin. is usually written as .
It effectively allows us to treat a point a vector.
! If and are the position vectors of and , then
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The distance of from the origin is:
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The distance between two points is:
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Basics #7: Writing Cartesian componentsAll 2D coordinates can be described by ‘some amount of ’ and ‘some amount of ’.We can use and as unit vectors pointing in the and directions.
𝒋=(01)
𝒊=(10)
!
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Given that and , find the exact value of .
Bro Tip: I really hate this notation, and just immediately write any vector in the usual form before solving more complex questions.
2𝒂+𝒃=( 8−2)?
Test Your Understanding
Straight Lines
𝑂
𝐴 𝐵𝑏
𝑎
Suppose we wanted to find an expression that represents any possible point on the line which passes through (with position vector ) and is parallel to .
This position vector? ? This? ?
This? ?
So a position vector describing all possible points?! The position vector of a line which passes through the point with position vector , and is parallel to , is:
where is a scalar parameter.We can represent it as a single vector
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Straight lines between two points
𝑂
𝐴 𝐵
𝑏𝑎
Now suppose that we only know the position vectors of two points, and . What is the equation of the line now?
! Equation of line passing through and :
(or )?
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Examples
Find the equation of the straight line which passes through and
𝑟=( 45−1)+𝑡(2−23 )
The straight line has vector equation . Given that lies on , find the value of and of .
With many straight line problems the strategy is to first find .Write line as single vector:
Show that and intersect and find the point of intersection.
Solve and components simultaneously:
You must check that the values match:
So they do intersect.Find point of intersection:
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Test Your Understanding
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Exam TopicsI went through a large number of C4 exams and looked at the vectors questions and their subparts. I found the following topics:
Topic Frequency
Points of intersection/showing lines do or don’t meet. 6
Angle between two lines 4
Finding length of vectors/distance between two points 4
Finding missing components of points on a line. 2
Finding nearest point on a line to the origin. 2
Finding nearest point on a line to a general point. 2
Dealing with perpendicular lines/showing lines a perpendicular 2
Finding angles between general vectors 1
Find missing point in a parallelogram 1
Show a point lies on a line 1
Show 3 points are collinear 1
Find the area of a rectangle formed by vectors 1
Find the area of a triangle formed by vectors 1
Find equation of a line 1
Reflection of a point in a line 1
Dot Product
The dot product of two vectors is the sum of the products of the components.
(321)⋅(401)=13 ( 5−20 )⋅(331)=9
(𝑎51 )⋅(3𝑏10)=3 𝑎+5𝑏+10
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Angle between two vectorsRemarkably, if the two vectors are unit vectors, the dot product gives us the cosine of the angle between them.
𝜃
! Angle between vectors:
but use:
𝒂
𝒃
Find the acute angle between the vectors and .
𝒂 ⋅𝒃=5+0+5=10?
Perpendicular vectors
! If two vectors are perpendicular then:
𝒂
𝒃
Using the equation from the previous slide…
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Show that and are perpendicular.
𝑎⋅𝑏= (2×1 )+(3×0 )+(1×−2 )=0?
Angles between straight lines
𝜃
𝒅𝟏
𝒂𝟐
𝑂
𝒂𝟏
𝒅𝟐
! If two lines have equations:
Then to find angle between them:
i.e. we only care about the directional part of the line, not how we got to the line.
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( 1−22 )𝒓=( 26−1)+𝑡(
1−22 )
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Test Your Understanding
( 21−1)⋅(3−15 )= (2×3 )+(1×−1 )+ (−1×5 )
�⃗�𝐵=(335)𝒓=( 2−15 )+𝑡(335)
Area
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Exam Topics
Topic Frequency
Points of intersection/showing lines do or don’t meet. 6
Angle between two lines 4
Finding length of vectors/distance between two points 4
Finding missing components of points on a line. 2
Finding nearest point on a line to the origin. 2
Finding nearest point on a line to a general point. 2
Dealing with perpendicular lines/showing lines a perpendicular 2
Finding angles between general vectors 1
Find missing point in a parallelogram 1
Show a point lies on a line 1
Show 3 points are collinear 1
Find the area of a rectangle formed by vectors 1
Find the area of a triangle formed by vectors 1
Find equation of a line 1
Reflection of a point in a line 1
Finding nearest point on a line to a pointThis one is not in your textbook!
( 1−23 )
𝑂
𝑃
𝑄
A straight line has the vector equation where is a scalar parameter. has the position vector . Find the point on the line which is closest to .
What do you notice about the special relationship between two of the vectors here?Direction of line and are perpendicular, i.e.
�⃗�𝑃=( 1+𝜆−2𝜆1+3𝜆)�⃗�𝑄=( 52−1)−(1+2𝜆−2𝜆
1+3 𝜆)=( 4−2𝜆2+2𝜆−2−3𝜆)
( 1−23 )⋅( 4−2𝜆2+2𝜆−2−3𝜆)=4−2𝜆−4−4𝜆−6−9𝜆=0
𝑃=( 35 , 45 ,− 15 )
(101)?
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Test Your Understanding
C4 Jan 2013 Q7
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