mapping with vectors vectors, who needs them ? how can they be useful to a geologist ? the flow of...
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Mapping With Vectors
Vectors, Who needs them ? How can they be useful to a Geologist ?
The flow of stream water can be described by vectorsindicating flow velocity and direction.
Let's Draw a Few Vectors
1. Vector a with length 3 cm and direction 10o CW from x-axis. (CW is “clockwise”)
2. Vector b with length 5 cm and direction 50o CCW from x-axis. (CW is “counter-clockwise”)
3. Vector c with length 3 cm and direction 190o CW from x-axis.
4. Draw vector d = a + b
5. Draw vector e = a + c
x
y
Mapping With Vectors
The continental divide describes drainage basins and water flowpatterns in North America.
Stream flow directions can be described by vectors on a map
Mapping With Vectors
The Earth's magnetic field shows specific directions formagnetic field lines.
To study magnetic reversals, colors and lines describe flow directions.
Mapping With Vectors
If you were hiking in the Sierra Nevada's and got lost lookingat your topo map (and your GPS batteries were dead), how could you find yourself ?
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Mapping With Vectors
How would you find the distance between two points on a globe ?
x
x
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Mapping With Vectors
What does an arrow tell you ?
Direction
Magnitude – That is, How fast ? How strong ?
Anything else ?
The length of an arrow can indicate field strength
Mapping With Vectors
What is the difference between a vector and a scalar ?
A scalar has a magnitude, but no direction (e.g temperature)
The velocity of this stream may be faster than this one.Does the direction matter ?
A Few ExamplesScalars or Vectors ?
Mass
Gravitational acceleration
Age
Depth of an aquifer
Concentration of a contaminant in ground water
Fault displacement
Seismic raypath
Magnitude of an earthquake
Source mechanism of an earthquake
Vector Fields
Wind direction over Bermuda and south Atlantic
Vectors have direction and length indicating flow
direction and wind strength
Vector Fields
Shear wave splitting measurements in western North America Azimuth indicates anisotropic fast direction Length of bars indicate strength of anisotropy
Mantle Flow directions can be predicted from global circulation models Predictions are compared to splitting observations to understand tectonic plate motion and flow in the upper mantle
Becker et al., 2006
How Do We Combine Vectors ?
Vectors show velocity and direction of stream flow. To determine the total flow direction from start to finish
you can add vectors
How Do We Combine Vectors ?
Vectors can be added together (they are commutative)
Just start the tail of one next to the head of another.
Vectors are denoted with a bar over the letter or a bold letter
ab
c
a + b = c
How Do We Combine Vectors ?
Vectors can be added together Vectors can be added in any order and will give the
same resulting vector
How Do We Combine Vectors ?
Vectors can be added in any order and will give the same resulting vector
How Do We Combine Vectors ?
Vectors can be added in any order and will give the same resulting vector
Let's Draw a Few Vectors
1. Vector a with length 3 cm and direction 10o CW from x-axis. (CW is “clockwise”)
2. Vector b with length 5 cm and direction 50o CCW from x-axis. (CW is “counter-clockwise”)
3. Vector c with length 3 cm and direction 190o CW from x-axis.
4. Draw vector d = a + b
5. Draw vector e = a + c
x
y
Which Way Did He Go ?
Geologists need to know directions!
Where are we ? Where are we going ?
Location of field sites, rock samples, instruments
Thinking about location in a quantitative way brings us
to the need to understand direction
Components of a Vector
A single vector can be broken down into two orthogonal vectors in a Cartesian reference frame
X (i)
Y (j)
r
r = 250 i + 145 j
Here, the term 250 i is a vector with magnitude 250 pointing in the i direction
250
145
Components of a Vector
Since directions i and j are at right angles to each other we can write this in another way using the Pythagorean Theorem
This will give us the length of the vector r, length is noted by vertical bars
X (i)
Y (j)
r
250
145
r = sqrt ( 2502 + 1452 )
Can We Obtain the Direction of the Vector ?
We can use rules for right triangles to obtain the angle
250
145
= arctan ( y / x) = arctan ( 145 / 250 ) = 30o
Convert from Polar to Cartesian Coordinates
You are facing east and see a lighthouse 30o to your right You walk to the lighthouse and determine it is 10 meters from
your original location This gives you the polar coordinates of the lighthouse in
your reference frame, where r = 10 m and = 30o
E
N
r
Convert from Polar to Cartesian Coordinates
You notice a river bed in your way and must go a longer route What is the distance if you go directly East, then South ? To convert polar coordinates to Cartesian coordinates,
x = r cos y = r sin
E
N
Convert from Cartesian to Polar Coordinates
You can also go in the reverse direction
E
N
r = sqrt ( x2 + y2 )
= arctan ( y / x )
Finding Yourself on a Map
If you were hiking in the Sierra Nevada's and got lost lookingat your topo map (and your GPS batteries were dead), how could you find yourself ?
?
Finding Yourself on a Map
1. Find 2 landmarks2. Find direction of where you are to these 2 landmarks3. Draw a line through each landmark4. The intersection of these lines is where you are on the map
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