How Do I Use Vectors?
Plotting And Calculating Magnitude and Direction in the Earth Sciences
Wind speed around the world can be represented as vectors. The length of the arrows indicates the speed and the orientation indicates the direction.
![[creative commons]](/images/creativecommons_16.png)
Provenance: EarthScope Consortium, created by Gabrielle Troia.
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
What Are Vectors and What Are They Good For?
To describe the temperature, it's enough to say it's 65 degrees F. If only a number and units are needed, we call this a scalar quantity. But direction also matters with something that is moving or a force causing motion. For example, wind moving at 11 meters per second (m/s) due south. If the description requires both an amount and a direction, it is a vector quantity. Vector quantities are often drawn on maps as arrows. In this module, vector quantities are shown in bold; whereas numerical-only, or scalar, quantities are in regular font.
How Do I Read a Diagram or Map that has Vectors?
Typically, on a map or diagram with vectors, there are one or more arrows. The length of an arrow is the magnitude, and the arrow points in the direction of the vector. Usually the tail end of the arrow marks the point that the vector describes.
Reading maps with vectors
GPS instruments use satellite data to measure the local velocity of the tectonic plate they are on. In this example, use the map to the right (click to enlarge) showing GPS velocity vectors to identify how fast a location is moving and in what direction. Each of the red arrows is a vector. This will show where there are active plate boundaries.
Vectors have both a
magnitude (a number) and a
direction. How do you tell the magnitude and direction of a vector on this map?
- What is the magnitude? In this case, the rate of motion of the GPS station in millimeters per year (mm/yr).
The length of the arrows corresponds to the magnitude of the velocity. In this discussion we use the word rate for the magnitude of the velocity. Note the scale in the lower left of the plate motion map, showing the length of the arrow that represents a rate of 20 mm/yr. An arrow twice as long would indicate a rate of 40 mm/yr.
- The direction of the vector is the orientation of the arrow.
For example, the fastest-moving point on this map, shown by the longest arrow in the lower right, is moving with an azimuth of 36o.
- On vector maps, usually the end of the tail of the arrow marks the point that the vector describes.
For example, the longest arrow in the lower right shows the velocity of the point at the tail of the arrow, the Cocos Islands.
Azimuth: In this module we describe map directions using
azimuth.
Azimuth depicting the relationship between positive and negative azimuth directions.
Provenance: Isabella Metts, EarthScope Student Career Intern
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Positive azimuth directions are defined in degrees counted
clockwise from north, as shown:
A negative azimuth is counted counterclockwise from north.
For example, = -26o is the same direction as 334o.
GPS station on Cocos Island near Costa Rica.
Provenance: https://www.unavco.org/instrumentation/networks/status/nota/photos/ISCO
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Estimate the rate of motion of the GPS receiver on the Cocos Islands (the longest arrow in the lower right of the map)?
You can estimate your answer. Compare visually the length of the Cocos arrow with the length of the scale arrow (lower left of the map). Longer arrows are moving at a faster rate. The Cocos arrow looks about 5 times as long as the 20 mm/yr scale vector, so the rate should be about `5 xx 20` mm/yr `= 100` mm/yr.
Identify trends in the vector map. Are there zones of greater/lesser plate motion magnitudes (rates)? Are there zones of distinctive directions of plate motions? Are there outlier points?
This map assumes the interior of the North American Plate is stable. There is no "perfect" answer to these interpretation questions. You could note that rates in the middle and eastern United States are very slow (short arrows). Rates on the west coast of the United States in California are much faster (long arrows), and consistently northwest in direction. The magnitudes and directions in Alaska are generally northward, but more variable than in California. The rate of motion of the Cocos Island GPS station is distinctly faster than anywhere else on the map, an outlier point.
How Do I Distinguish Vectors from Scalars?
Map showing the plate motion vectors as measured by the GPS stations in the Network of the Americas (NOTA).
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Vectors have both a
magnitude (a number) and a
direction, while scalars only have a magnitude.
Map of GPS velocities
The map of GPS velocities shows two types of data: elevations (shown by color) and plate motion velocities (shown with red arrows). Determine which of these two types of data (elevation, velocity) is a scalar and which is a vector.
Vector-or-scalar Step 1. Determine if the value has a direction and a value or only a numeric value.
Elevations are shown in colors from darker blue at lowest seafloor, yellow and brown at high land elevations. Elevation is a property that has a numerical value, such as 500 feet above sea level, but elevation does not have a direction.
Plate motion velocities have both a magnitude (mm/yr) and a direction.
Vector-or-scalar Step 2. If the data has a numeric value only, it is a scalar value. If it has a magnitude and direction, it is a vector.
Since the elevation does not have a direction, it is a scalar. Plate motion velocities have both a magnitude (mm/yr) and a direction, so they are vectors.
Bold text, such as ` bb"M"`, is used to show a vector. Sometimes subscripts are added to differentiate between different vectors of the same type. For example ` bb"M"_(bb"x") ` or ` bb"M"_(bb"y") ` may be used to represent vectors of the same type, with different magnitudes or directions.
How Do I Multiply a Vector by a Scalar?
Multiplying by a positive scalar results in a vector with a different magnitude but the same direction. When multiplying a vector by a negative scalar the magnitude changes and the direction flips 180 degrees.
Provenance: https://en.wikipedia.org/wiki/Volcanic_bomb
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Velocity of a volcanic bomb
Explosive volcanic eruptions are dynamic events where material is ejected from the volcanic conduit at high velocities. Large (> 2 mm) projectiles are called "bombs" and can fall kilometers away from the vent.
A volcanic bomb is erupted from a volcano with an initial upward velocity ( `bb"V"_(bb"up")`) of 200 meters per second (m/s). The downward velocity ( `bb"V"_(bb"down")`) of the bomb just before impact with the ground is 1/4 of the initial velocity in the opposite direction. What is the velocity of the bomb just before impact?
Multiply-by-scalar Step 1. Identify the magnitude and direction of the vector to be multiplied and the scalar multiplication factor.
Vector diagram illustrating the upward velocity of a volcanic bomb.
![[reuse info]](/images/information_16.png)
Provenance: John Zayac, Vassar College
Reuse: This item is in the public domain and maybe reused freely without restriction.
The initial upward velocity of the bomb is 200 m/s in an upward direction. Note that for this problem, positive velocities are in the upward direction.
` bb"V"_(bb"up") = 200` m/s
The scalar multiplication factor is given as 1/4. Still, because it is in the opposite direction, we need to multiply by a negative multiplication factor in order to account for the directional change. So the scalar multiplication factor is -1/4 or -0.25.
Multiply-by-scalar Step 2. Multiply the vector magnitude by the scalar to get the new magnitude and direction.
Vector diagram showing the relationship between vectors in the problem.
Provenance: John Zayac, Vassar College
Reuse: This item is in the public domain and maybe reused freely without restriction.
The multiplication factor adjusts the magnitude of our downward velocity vector but not the direction. Here, the given velocity (
`bb"V"_(bb"up")`) has an upward direction and
`bb"V"_(bb"down")` has the opposite direction.
` (bb"V"_(bb"down")) = -0.25 * (bb"V"_(bb"up")) = -0.25 * 200` m/s` = -50` m/s.
Multiply-by-scalar Step 3. Does your answer make sense?
-50 m/s is 1/4 of the original velocity and in the opposite direction, so our math seems to be correct.
How Do I Find the Components of a Vector?
Vectors can be thought of as a sum of two perpendicular components. One way to think of these components is that they represent parts of the force/flow/velocity vector operating in specific directions. The original vector is called the resultant of the two components.
Finding vector components requires trigonometry. If you are struggling, here are some tips:
- You can review Trigonometry.
- Make sure you know whether your calculator expects angles in degrees or radians. Get help using your calculator.
- The components of a vector are always smaller than the resultant. Often the components and resultant make a right triangle with the resultant as the hypotenuse.
- Note: Excel in default mode uses angles in radians. `1 "radian" = "degrees" * pi/180` . In Excel `pi` is written as pi().
Resolving a plate motion vector into orthogonal components
The Cocos Islands station velocity vector (`bb"V"_bb"C"`) has a magnitude (rate) of 116 mm/yr and an azimuth of 36o. What is the eastward component of the rate of motion of the Cocos Island GPS station? What is the northward component of the rate of motion?
Components Step 1. Draw a diagram that shows the resultant vector magnitude and direction, and the direction of the components you want to solve for.
Since the total velocity `= bb"V"_bb"C"= 116` mm/yr, this is the resultant vector. Its direction is 36° with a length of 116 mm, so sketch this as an arrow. Then show the components you are asked for (northward and eastward).
A map of the Cocos Island GPS vector (Vc) in red with the directions North and East labled with black dashed arrows
Provenance: Eric Baer, Highline College
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Click on the image to enlarge.
Components Step 2. Find the right triangle that includes the resultant, the components and angle.
Map of cocos island GPS vector and the directions of the components with a right triangle drawn in.
Provenance: Eric Baer, Highline College
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
The purple triangle on the vector component figure to the right includes the resultant, each of the components, and angle. Note that the top side of the triangle is the same length as,
`bb"V"_(bb"C(E)")`. So the length of the top side of the triangle is the magnitude
`bb"V"_(bb"C(E)")`.
Components Step 3. Use trigonometry to solve for the lengths of each component.
Find the length of the top side of the purple triangle since this is the magnitude of the east component
`bb"V"_(bb"C(E)")`.
Using trigonometry,
- `sin(alpha) = (opp)/(hyp) = bb"V"_(bb"C"(bb"E"))/bb"V"_bb"C"` .
Rearranging,
- `bb"V"_(bb"C"(bb"E"))` = `sin(alpha)* bb"V"_bb"C"`
Substituting in values,
- `= sin(36) * 116` mm/yr `= 0.59 * 116` mm/yr = 68 mm/yr .
So the east component has a magnitude of 68 mm/yr.
Then find the length of the vertical side of the triangle, since this is the magnitude of the north component `bb"V"_(bb"C(N)")`.
Using trigonometry,
- `cos(alpha) = (adj)/(hyp) = bb"V"_(bb"C"(bb"N"))/bb"V"_bb"C"` .
Rearranging,
- `bb"V"_(bb"C"(bb"N"))= cos(alpha)*bb"V"_bb"C"`
Substituting in values,
- `cos(36)* 116` mm/yr `= 0.81 * 116` mm/yr` = 94` mm/yr .
So the north component has a magnitude of 94 mm/yr.
Does your answer make sense?
Check 1: Should the north and east components of motion have faster or slower rates than the resultant `bb"V"_(bb"C")`? Component magnitudes will always be smaller than the resultant magnitude, as shown by the blue triangle. 68 mm/yr eastward and 94 mm/yr northward are indeed slower than the resultant 116 mm/yr.
Check 2: Just looking at the direction of `bb"V"_(bb"C")`, should the northern component be greater (faster) than the eastern? Yes, from the arrow direction the plate is moving more northerly than easterly. 94 mm/yr northward is more than 68 mm/yr eastward.
Check 3. The components and the resultant vector should work in the Pythagorean theorem. Does 1162=682+942? They are very close, with a small difference because of rounding.
How do I find vector components relative to a given direction?
Provenance: https://upload.wikimedia.org/wikipedia/commons/b/b2/Landslide_in_Cusco%2C_Peru_-_2018.jpg Attribution Galeria del Ministerio de Defensa del Perú, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Some problems require finding vector components perpendicular and parallel to a given direction. This involves the same steps as finding east and north components of a map vector, but it may be helpful to visually rotate your problem, so your given direction is either horizontal or vertical.
Landslide forces
Intuitively, we would guess that the steeper the dip of the slip surface, the more likely it is that a landslide would slip. This problem asks you to check that intuition using the physics of slope stability.
The forces driving and resisting landslide slip depend on the gravitational force `bb"G"` on the landslide and the dip angle of the rupture surface theta as shown in Figure AA. The gravitational force `bb"G"` can be resolved into the downslope component `bb"G"_(bb"d")`, parallel to the rupture surface and the perpendicular component `bb"G"_(bb"p")`. `bb"G"_(bb"d")` drives slip of the body of the landslide down the rupture surface. In contrast, the perpendicular component `bb"G"_(bb"p")` helps the landslide 'stick' to the underlying rock, so it resists slip.
Compare rupture surface dip angles of 70o and 40o. Show that for the steeper 70o dip angle, the downslope force `bb"G"_(bb"d")` is greater and the resisting force `bb"G"_(bb"p")` is weaker than the 40o scenario. Assume the magnitude of the gravitational force of the body of the landslide `bb"G"` is 1.5 x 1010 N.
Figure AA. Forces driving a landslide can be represented as vectors.
Provenance: EarthScope Consortium, Gabrielle Troia.
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
To solve this we will follow the steps for finding the components of vectors in a given direction for both the 40o degree and 70o dip angle cases.
Components Step 1. Draw a diagram that shows the resultant vector magnitude and direction, and the direction of the components you want to solve for.
In this case, the drawing is shown in figure AA
Figure AA. Forces driving a landslide can be represented as vectors.
Provenance: EarthScope Consortium, Gabrielle Troia.
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Components Step 2. Find the right triangle that includes the resultant, the components, and angle.
Figure AB. Triangle used to find magnitude of component vectors.
Provenance: Sarah Kruse, University of South Florida
Reuse: This item is in the public domain and maybe reused freely without restriction.
This diagram shows the right triangle (in yellow) that includes the components, and the angle
`theta` which is the steepness of the slope. Note that the angle
`theta` in the triangle is equal to the steepness of the slope because they are both
complementary two angles whose measurements add to 90 degrees angles to the angle between
`bb"G"` and
`bb"G"_(bb"d")`.
In the question, it states that the resultant vector magnitude = `bb"G" = 1.5 xx 10^(10) N`.
For this problem, you will need to solve for `theta` values of 40o and 70o.
Components Step 3. Use trigonometry to solve for the lengths of each component.
Figure AB. Triangle used to find magnitude of component vectors.
Provenance: Sarah Kruse, University of South Florida
Reuse: This item is in the public domain and maybe reused freely without restriction.
- `sin(theta) = (opp)/(hyp) = bb"G"_bb"d"/bb"G"`. Rearranging, `bb"G"_bb"d" = sin(theta)* bb"G"`
- For `theta = 40`o `bb"G"_bb"d"= 0.64 * 1.5 xx 10^(10) N = 9.6 xx 10^9 N`.
- For `theta = 70`o ` bb"G"_bb"d"= 0.94 * 1.5 xx 10^(10) N = 1.4 xx 10^(10) N`.
- ` cos(alpha) = (adj)/(hyp) = bb"G"_bb"p"/bb"G"`. Rearranging, ` bb"G"_bb"p" = cos(alpha) * bb"G"`
- For `theta = 40`o `bb"G"_bb"p" = 0.77 * 1.5 xx 10^(10) N = 1.1 xx 10^(10) N`.
- For ` theta = 70`o ` bb"G"_bb"p" = 0.34 * 1.5 xx 10^(10)N = 5.1 xx 10^9 N`.
Does your answer make sense?
The answers in Step 3 show that for the steeper 70o dip case, the downslope force `bb"G"_(bb"d")` is greater, and the perpendicular force `bb"G"_(bb"p")` is smaller. This means the force driving slip is greater, the force acting to restrain slip is smaller. So the landslide is more likely to slip at the steeper dip angle, as matches intuition.
How Do I Add Vectors?
Two vectors are added by resolving them into the same directional components, adding the magnitudes of the parallel components and then adding the resulting non-parallel components using trigonometry and the Pythagorean theorem.
Vector addition can be visualized graphically. When two vectors are being added, the sum is the vector that connects the two when the arrowhead of one vector is connected to the tail of the other.
Provenance: https://en.wikipedia.org/wiki/File:Vector_Addition.svg
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Hot spot velocities
The volcanoes of the Hawaiian Emperor Seamount Chain provide a record of the motion of the Pacific Plate over the Hawaiian hotspot for the past 80 million years (Myr). The plate has been moving to the northwest for the past 47 Myr, but was traveling more northerly before that. The distance between the Big Island of Hawaii, the current location of the hotspot (age = 0 Myr), and the Daikakuji Seamount (age = 47 Myr) is 3520 km with an azimuth of 300o. Between Daikakuji Seamount and Meiji Seamount (age = 82 Myr) the distance is 2375 km with an azimuth of 351o.
What is the net velocity of the Pacific Plate over the past 80 Myr?
Map illustrating the motion vectors of the Hawaiian-Emperor Seamount Chain with an inset that displays the vector components.
Provenance: John Zayac, Vassar College
Reuse: This item is in the public domain and maybe reused freely without restriction.
Add Vectors Step 1. Identify the magnitude and direction of each vector given in the problem, and determine which vector needs to be solved for.
The rate of movement here is simply the distance divided by the time, so we will use the formula: rate = distance/(age
old - age
young)
For the Hawaii to Daikakuji segment, the rate `(r_(H-D)) = (3520 (km)) /(47 (Myr) - 0 (Myr)) = 75 `km/Myr` = 75 `mm/yr``
For the Daikakuji to Meiji segment, the rate `(r_(D-M)) = (2375 (km))/(82 (Myr) - 47 (Myr)) = 68 `km/Myr` = 68 `mm/yr``
These values represent the magnitudes of our vectors.
The directions are given in the question as azimuths of 300o and 351o respectively.
Add Vectors Step 2. Break down each of the given vectors into their respective directional components.
Recall that your east components will be negative since the plate is moving westward.
The components can be obtained using the following relations (as in the map components example above):
Hawaii to Daikakuji segment:
`bb"V"_(bb"H-D"(bb"N")) = cos(alpha) * bb"V"_(bb"H-D") = cos(300`o`) * 75` mm/yr` = 0.500 * 75 `mm/yr` = 38 `mm/yr
`bb"V"_(bb"H-D"(bb"E")) = sin(alpha) * bb"V"_(bb"H-D") = sin(300`o`) * 75 `mm/yr` = -0.866 * 75 `mm/yr` = -65 `mm/yr
Daikakuji to Meiji segment:
`bb"V"_(bb"D-M"(bb"N"))= cos(alpha) * bb"V"_(bb"D-M") = cos(351`o`) * 68 `mm/yr` = 0.988 * 68 `mm/yr` = 67 `mm/yr
`bb"V"_(bb"D-M"(bb"E"))= sin(alpha) * bb"V"_(bb"D-M") = sin(351`o`) * 68 `mm/yr` = -0.156 * 68 `mm/yr` = -11 `mm/yr
Add Vectors Step 3. Add the magnitudes of the component vectors with the same direction to calculate the component vectors of the sum.
Add the north components of both segments to get the north component of the resultant vector.
` bb"V"_(bb"total"(bb"N")) = bb"V"_(bb"H-D"(bb"N")) + bb"V"_(bb"D-M"(bb"N")) = 38 `mm/yr` + 67 `mm/yr` = 105 `mm/yr
Add the east components of both segments to get the east component of the resultant vector.
` bb"V"_(bb"total"(bb"E")) = bb"V"_(bb"H-D"(bb"E")) + bb"V"_(bb"D-M"(bb"E")) = -65 `mm/yr` + -11 `mm/yr` = -76 `mm/yr
Add Vectors Step 4. Use your results in Step 3 to find the magnitude and azimuth of the sum vector.
Recall that the component vectors are perpendicular to each other, so they form the opposite and adjacent sides of a right triangle. This means that the magnitude of the resultant vector can be found using the Pythagorean theorem:
`(bb"V"_(bb"total"))= sqrt[ (bb"V"_((bb"total"(bb"N"))))^bb"2" + (bb"V"_((bb"total"(bb"E"))))^bb"2"] = sqrt((105 "mm/yr")^2 + (-76 "mm/yr")^2) = 130` mm/yr
The azimuth can be determined by using the relation:
`alpha = tan^(-1) ((bb"V"_(bb"total"(bb"E")))/(bb"V"_(bb"total"(bb"N")))) = tan^(-1) (-76 `mm/yr` / 105 `mm/yr`) = -36`o `
Remember if your azimuth calculation results in a negative number, you can convert that to a positive azimuth by subtracting it from 360o.
The net velocity of the plate has been 130 mm/yr with an azimuth of 324o.
Add Vectors Step 5. Does your answer make sense?
Check the magnitude: Your resultant vector should be longer than the original vectors and shorter than if you put the vectors in a line from head to tail. So the magnitude of the sum should be greater than 68 mm/yr and 75 mm/yr and less than 143 mm/yr. The magnitude is 130 mm/yr, so the magnitude of the answer makes sense.
Check the azimuth: Assess visually from the vector diagram. The azimuth should be between the azimuths of the original velocity vectors (300o and 351o). The calculated azimuth is 324o so the result seems consistent.
Where Are Vectors Useful in the Earth Sciences?
Vectors are important to many subdisciplines of the Earth sciences. A few examples
Meanings and maps of vectors:
- Global Positioning System (GPS) measurements of plate motions
- The movement of contaminant plumes in the atmosphere, oceans, and freshwater systems
One-dimensional vector addition/subtraction and scalar multiplication:
- Calculating effective sea level rise with both rising sea level and subsiding land
- Examining the forces driving sinkhole collapse
- Tracking the rise of bubbles and sinking of crystals in magmatic systems
Two-dimensional vector components and addition/subtraction:
- Finding the relative velocity of two tectonic plates along their boundary
- Tracking the motion of sand along a coastline due to longshore transport
- Computing groundwater flow rate and direction from three wells
- Applying Snell's Law to wave refraction and groundwater flow refraction
- Examining the forces involved in slope stability/failure in landslides
- Understanding the readings of commonly-used magnetometers that measure the magnetic field strength (but not the magnetic field direction)
Next steps
I am ready to PRACTICE!
If you think you have a handle on the steps above, click on this bar to try practice problems with worked answers.
Or, if you want even more practice, see 'More help' below.More Help (Resources for Students)
Pages written by Sarah Kruse, University of South Florida, and John Zayac, Vassar College.
Show me more
Hide