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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rsMathematical Interlude 1Spaces, Trigonometry, and Vectors
1
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Spatial Coordinates
A spatial coordinate is an• ordered• tuple• of one or more real numbers.• (-14.2, 6.0, 23.1)
Coordinates define position in a coordinate system, or space.
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
• 1637 by Rene Descartes• Specified by three, orthogonal vectors,
each of length one, and an origin.• Usually shown as
Cartesian Coordinate System
3x
y
z
origin
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
More Coordinates
• The order of (x, y, z) is important.• The spatial coordinates have the
dimensions of length.
• We add a positive-going time coordinate to make a reference frame. We assume• time flows uniformly, and• times can be compared at different
locations. 4
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Even More Coordinates
We’ll have more to say about coordinates when we reach the topic of vectors, later in this segment.
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Trigonometry: Degrees and Radians
6
One revolution equals 360 degrees.
One revolution equals 2 radians.
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Trigonometric Functions
7
(radians)
hypotenuse = h
oppo
site
= o
adjacent = a
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Trigonometric Functions (cont.)
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hypotenuse = h
oppo
site
= o
adjacent = a
sin (𝜃 )=𝑜h=
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos (𝜃 )=𝑎h=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tan (𝜃 )=𝑜𝑎
=𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
(soh)
(cah)
(toa)
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Trigonometric Functions (cont.)Mnemonic:
Chief Soh-Cah-Toa
or, if you prefer,
Camp Soh-Cah-Toa9
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
sin() and cos()
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
tan() too
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Common Circular Motion
12
r𝑥=𝑟 cos (𝜃)
𝑦=𝑟 sin (𝜃)
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
More Trigonometry
• See the Wikipedia page for “Trigonometric Functions”• One simple and very useful relation is
• This is just
13
o2
h2 +a2
h2 = 1
o2 + a2 = h2
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Vectors
• A vector is a geometric object with• a length and• a direction.
• You can imagine it as an arrow floating in space that can be moved around freely.
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Representing Vectors
• If some symbol, say , represents a vector we often indicate that with an overhead arrow, like this .
• We use the absolute-value sign, , to represent the length of a vector:
15
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Representing Vectors (cont.)
• In the special case that
we call a unit vector.
• If is defined to be a unit vector
we sometimes express it as (we replace the arrow with a caret).
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
We can multiply a vector by a real number (often called a scalar). The vector points in the same direction as and is twice as long:
Vector Arithmetic (I)
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
• We can add two (or more) vectors together by joining them tip-to-tail to form a quadrilateral:
Vector Arithmetic (I) (cont.)
18
�⃗� �⃗�
�⃗�+�⃗�
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Vector Components
• We often represent a vector by its x, y, and z components.
• We define , , and to be unit vectors pointing in the x, y, and z directions, respectively.
• We specify by the three scalars, , , and , thus
19
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Vector Components (cont.)
Elementary geometry gives
20
x
y
z
origin
�̂�
�̂�
�̂�
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Products of Vectors
There are two distinct ways of forming the product of two vectors:
• one is called the dot product and returns a simple scalar:
• the other is called the cross product and returns a vector:
• for now all we care about is the dot product.
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Dot Product of Two Vectors
• The dot product is defined geometrically as where is the angle between and
• The equivalent definition in terms of the components of and is
22
Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Two Useful Aspects of the Dot Product
Dot product of a vector with itself:
Dot product of two orthogonal vectors:
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Spac
es, T
rigon
omet
ry, a
nd V
ecto
rs
Next Up
Calculus
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