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Basic vectors
SLIDE 1
Scalars and vectors
A scalar is a quantity with only size (magnitude).
A vector is a quantity with size and direction.
Question
The following is a list of pairs of quantities, one of which is a scalar and the other a vector. In each case, explain the difference and state which is the scalar and which the vector.
? distance and displacement
? speed and velocity
? temperature and increase of temperature
SLIDE 2
? distance ? scalar, displacement ? vector
Distance is the magnitude of the separation between two points A and B; it is the same regardless of the direction in which an object travels between the two points. Displacement is distance and direction combined.
? speed ? scalar, velocity ? vector
Speed is the rate of change of distance with time; velocity is rate of change of displacement with time.
? temperature ? scalar, increase of temperature ? vector
Temperature is a measure of the total (internal) energy of an object, relative to a zero. A scalar is usually a positive number (integer or real), but it can be negative, as it may depend on where the zero value is defined. For example, temperature is a scalar quantity, though it can be negative if the zero degrees is defined to be the melting point of water. An increase of temperature involves the direction of the increase – if negative, it is a decrease of temperature, so it is a vector.
SLIDE 3
The following is a list of pairs of quantities, one of which is a scalar and the other a vector. In each case, explain the difference and state which is the scalar and which the vector.
? force and mass
? position on the circumference of a circle, the length of the radius of a circle
? the movement from point A to B, the length of the line segment from A to B.
SLIDE 4
? force ? vector, mass ? scalar
Force is a vector. Forces cause objects to accelerate, and acceleration – change of velocity – is a vector, so force is also a vector. Mass is a measure of the quantity of substance and is a scalar.
? position on the circumference of a circle ? vector, the radius of the circle ? scalar
Position on a circle can change and move about, so it is a vector – you require two quantities to describe it, for example, an x and a y coordinate. The radius of a circle is the same for a given circle regardless of the position of a point on the circumference and is a scalar quantity.
? the movement from point A to B, the length of the line segment from A to B.
The movement from A to B has a direction, so is a vector. If you move from B to A this is the reverse of the movement from A to B. It is a displacement. The length of the line segment
SLIDE 5
and are two points in the xy-plane.
? Make a sketch of the xy-plane, showing A, B and the displacement from A to B. Mark this displacement .
? Find the horizontal displacement in the positive x-direction from A to B. Find also the vertical displacement in the positive y-direction from A to B. Show these displacements on your diagram.
? What is the relationship between and the displacement from B to A, ? What are the horizontal and vertical displacements of ?
? Find the distance .
? What is the relationship between and the distance ?
SLIDE 6
The horizontal displacement of is . The vertical displacement of is .
The horizontal displacement of is . The vertical displacement of is .
The distance is .
.
SLIDE 7
Vector component form
is the vector from A to B.
Here 3 is the component of in the positive x-direction, and 5 is the component of in the positive y-direction.
is a column vector in which the first (top) number is the component of the vector in the x-direction, and the second (bottom) number is the component of the vector in the y-direction.
SLIDE 8
What is the dimension of the above vector?
SLIDE 9
has two components and two dimensions. It is a 2-dimensional vector.
SLIDE 10
A 2-dimensional vector has two components. As a column vector it is written
where x is the component of the vector in the x-direction, and y is the component of the vector in the y-direction.
Question
How many components in a column vector does a 3-dimensional vector need?
SLIDE 11
A three-dimensional vector requires three components. The three-dimensional vector from A to B in three-dimensional space has the column vector .
SLIDE 12
? Make a sketch of the xy¬-plane showing the points , and .
? Onto the diagram mark the vectors , and . Find these vectors in component form.
? Find in component form the vectors , and .
? Find . How does your result correlate with your diagram?
SLIDE 13
Travelling around the triangle from A to B to C to A brings you back to the same place from whence you started.
SLIDE 14
and are two points in the xy¬-plane.
? Make a sketch showing A and B and the vector .
? In terms of their components, find the vector and the length .
? What difference, if any, is there between and
SLIDE 15
reads, “the magnitude of the vector ” while reads, “the length of the line segment AB”. These are both scalar quantities having the same magnitude, so represent a difference or nuance of meaning only. We write, . For all practical purposes, they are the same.
SLIDE 16
? Write each of the vectors , and in component form as column vectors.
? What do you observe about the relationship between these vectors. Are they different vectors or the same vector?
SLIDE 17
?
? All of these vectors have the same components. Because of this, they are regarded as the same vector. This is necessary, because otherwise, we would introduce a contradiction into vector algebra.
SLIDE 18
Identity of displacement vectors
Any two vectors with the same components are said to be the same displacement vector.
Question
What would happen if we did not adopt the rule that vectors with the same components are not identical?
Hint. Suppose above that (not equal). Show that this leads to a contradiction.
SLIDE 19
Suppose
Then .
This is a contradiction, so we are forced to make displacement vectors the same if they have the same components, regardless of where they “appear” in the xy¬-plane.
SLIDE 20
Displacement and position vectors
Although the displacement vectors are equal because they have the same components, the diagram indicates that in a different sense they different, because they start at different positions in the xy-plane, and connect different points.
Advanced
We have removed some of the information contained in the vectors when we write them in component form. We can reintroduce the information that we have lost by defining a position vector. This position vector gives the position of a point relative to the origin. The position vector of A is .
At this level, all vectors are displacement vectors, and they will be simply called vectors.
SLIDE 21
Abstract representation of a vector
A vector can be represented by a single letter.
a
Because a vector is a two or more-dimensional quantity, it is written in bold type in text books, and underlined in hand-written script.
State which of these vectors are equal and find the odd man out.
SLIDE 22
The odd man out is
SLIDE 23
Addition of vectors
When adding vectors, add the components separately
Example
SLIDE 24
Find
? ?
? ?
SLIDE 25
?
?
?
?
SLIDE 26
Find
SLIDE 27
SLIDE 28
? Find , , and
? What do you observe about and ?
SLIDE 29
?
?
SLIDE 30
We have and . By the identity of vectors , and .
Question
How many triangles are there, and do we need the coordinate grid?
SLIDE 31 – EXPLANATION AND SOLUTION
Removing the coordinate plane
Since and we can remove the coordinate grid. We can represent the vectors using abstract vectors, where , and .
SLIDE 32
Triangle law of addition
SLIDE 33
?
Find in terms of
?
Find in terms of
SLIDE 34
?
?
SLIDE 35
Anchoring vectors
All vectors follow the parallelogram rule on the left, where all displacements are thought of as starting at the origin.
But we use the triangle on the right because it is easier to picture the addition of vectors that way.
SLIDE 36
Vectors in three-dimensions
Find
? ?
? ?
SLIDE 37
?
?
?
?
SLIDE 38
Notation
the line segment from P to Q
the length of the line segment from P to Q; the distance from P to Q
the (displacement) vector from P to Q
the length (modulus) of the vector from P to Q.
Custom
It is customary to shorten this list
the line segment from P to Q and the distance between P and Q.
the (displacement) vector from P to Q
We use the modulus when we explicitly find the length of a vector.
SLIDE 39
Find
? ?
SLIDE 40
?
?
SLIDE 41
Find
? ?
SLIDE 42
Find
?
?
SLIDE 43
,
Find
? ? ?
SLIDE 44
,
?
?
?
SLIDE 45
Find in terms of x and y
? ?
? ?
SLIDE 46
? ?
? ?
SLIDE 47
M is the mid-point of AC. Find
? ?
? ?
SLIDE 48
M is the mid-point of AC.
? ?
?
?
SLIDE 49
M is the mid-point of BC. Find
?
?
?
SLIDE 50
M is the mid-point of BC.
?
?
?
SLIDE 51
i,j – notation
This is a way of writing a vector in a single line
i is a unit vector in the x-direction, and j is a unit vector in the y-direction.
SLIDE 52
? Write , , in i, j form.
? Convert , and to column vector form.
SLIDE 53
?
?
SLIDE 54
, and . Find
?
?
?
SLIDE 55
, and .
?
?
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