1. Mapping With Vectors Vectors, Who needs them ? How can they be useful to a Geologist ? The flow of stream water can be described by vectors indicating flow velocity and direction.

2. Let's Draw a Few Vectors 1. Vector a with length 3 cm and direction 10 o CW from x-axis. (CW is “clockwise”) 2. Vector b with length 5 cm and direction 50 o CCW from x-axis. (CW is “counter-clockwise”) 3. Vector c with length 3 cm and direction 190 o CW from x-axis. 4. Draw vector d = a + b 5. Draw vector e = a + c x y

3. Mapping With Vectors The continental divide describes drainage basins and water flow patterns in North America. Stream flow directions can be described by vectors on a map

4. Mapping With Vectors The Earth's magnetic field shows specific directions for magnetic field lines. To study magnetic reversals, colors and lines describe flow directions.

5. Mapping With Vectors If you were hiking in the Sierra Nevada's and got lost looking at your topo map (and your GPS batteries were dead), how could you find yourself ? ?

6. Mapping With Vectors How would you find the distance between two points on a globe ? x x ?

7. 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

8. 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 ?

9. A Few Examples Scalars 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

10. Vector Fields Wind direction over Bermuda and south Atlantic Vectors have direction and length indicating flow direction and wind strength

11. 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

12. 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

13. 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 a b c a + b = c

14. How Do We Combine Vectors ? Vectors can be added together Vectors can be added in any order and will give the same resulting vector

15. How Do We Combine Vectors ? Vectors can be added in any order and will give the same resulting vector

16. How Do We Combine Vectors ? Vectors can be added in any order and will give the same resulting vector

17. Let's Draw a Few Vectors 1. Vector a with length 3 cm and direction 10 o CW from x-axis. (CW is “clockwise”) 2. Vector b with length 5 cm and direction 50 o CCW from x-axis. (CW is “counter-clockwise”) 3. Vector c with length 3 cm and direction 190 o CW from x-axis. 4. Draw vector d = a + b 5. Draw vector e = a + c x y

18. 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

19. 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

20. 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 25 0 14 5 r = sqrt ( 250 2 + 145 2 )

21. Can We Obtain the Direction of the Vector ? We can use rules for right triangles to obtain the angle 25 0 14 5 q = arctan ( y / x) = arctan ( 145 / 250 ) = 30 o q

22. Convert from Polar to Cartesian Coordinates q You are facing east and see a lighthouse 30 o 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 q = 30 o E N r

23. 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 q y = r sin q E N

24. Convert from Cartesian to Polar Coordinates You can also go in the reverse direction E N r = sqrt ( x 2 + y 2 ) q = arctan ( y / x )

25. Finding Yourself on a Map If you were hiking in the Sierra Nevada's and got lost looking at your topo map (and your GPS batteries were dead), how could you find yourself ? ?

26. Finding Yourself on a Map 1. Find 2 landmarks 2. Find direction of where you are to these 2 landmarks 3. Draw a line through each landmark 4. The intersection of these lines is where you are on the map ?