Spac

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rsMathematical Interlude 1Spaces, Trigonometry, and Vectors

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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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• 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

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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

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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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Trigonometry: Degrees and Radians

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One revolution equals 360 degrees.

One revolution equals 2 radians.

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Trigonometric Functions

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(radians)

hypotenuse = h

oppo

site

= o

adjacent = a

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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)

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Trigonometric Functions (cont.)Mnemonic:

Chief Soh-Cah-Toa

or, if you prefer,

Camp Soh-Cah-Toa9

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sin() and cos()

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tan() too

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Common Circular Motion

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r𝑥=𝑟 cos (𝜃)

𝑦=𝑟 sin (𝜃)

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More Trigonometry

• See the Wikipedia page for “Trigonometric Functions”• One simple and very useful relation is

• This is just

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o2

h2 +a2

h2 = 1

o2 + a2 = h2

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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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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:

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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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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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• We can add two (or more) vectors together by joining them tip-to-tail to form a quadrilateral:

Vector Arithmetic (I) (cont.)

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�⃗� �⃗�

�⃗�+�⃗�

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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

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Vector Components (cont.)

Elementary geometry gives

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x

y

z

origin

�̂�

�̂�

�̂�

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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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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

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Two Useful Aspects of the Dot Product

Dot product of a vector with itself:

Dot product of two orthogonal vectors:

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Next Up

Calculus

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