Vectors - Practice Problems
Solving Earth science problems with direction and magnitude
This module is available for public use, but it is undergoing revision after classroom implementation with the Math Your Earth Science Majors Need project.
Examining relative sea level change in North America
Sea level is not changing at the same rate everywhere. Relative sea level change is the sum of sea level rising due to climate change and land moving up or down, in part due to the isostatic adjustment of the continent after ice sheet melting since the last glacial maximum. We can investigate the balance between these phenomena using a vector map of the rates of relative sea level change along the coastlines of North America.
Fig. 1. NOAA Sea Level Map
![[reuse info]](/images/information_16.png)
Provenance: https://tidesandcurrents.noaa.gov/sltrends/
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Problem 1: Use the map in Figure 1 to identify and interpret patterns in the rates of relative sea level change along the North American coastlines.
Problem 1a. This map shows both shaded topography and sea level change rates. Are each of the properties a scalar value or a vector?
Topography is the representation of elevation - which has a numerical value, but not a direction so it is a scalar value.
Sea level change has a numerical value (a magnitude) and a direction (up or down) so it is a vector.
Problem 1b. Look at the scale for magnitude of the vectors. What is the magnitude of the greatest rate of relative sea level rise vector on this map? What is the magnitude of the greatest rate of relative sea level fall vector represented in the map?
There is a vector along the Gulf Coast, near the Mississippi River delta, where the rate of relative sea level rise is greater than 9 millimeter per year (mm/yr). There are several vectors on the southern coast of Alaska and the western edge of Hudson Bay where the rate of relative sea level fall is more than 9 mm/yr.
Problem 1c. Observe the trends present in the map. Are there regions that have overall high relative sea level rise vectors? What about relative sea level fall?
The east coast and the Gulf coast appear to have the highest rates of relative sea level rise; the rise is slower along the west coast. Relative sea level is falling along the southern coast of Alaska.
Movement of tectonic plates
Fig. 2. India plate motions with vectors at 55 and 10 Ma
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Vector math is important to the study of the movement of tectonic plates because the vectors are the way scientists describe plate motion magnitude and direction.
Problem 2. 55 million years ago, India was moving northeastward toward Asia at a rate of 150 mm/yr (fast!). But as India neared Asia the rate slowed down, so that by 10 million years ago it was only moving one fourth as fast as it had been. The motion at 55 million years ago can be described by a vector ` bb"V"_(bb"55mya")` with magnitude 150 mm/yr and azimuth 14o. What are the magnitude and direction of the velocity vector ` bb"V"_(bb"10mya")` for motion 10 million years ago?
Step 1. Identify the magnitude and direction of the vector to be multiplied and the scalar multiplication factor.
Original vector magnitude = 150 mm/yr. Direction = azimuth 14o --> keep track of this, but it won't change. Scalar multiplication factor = one fourth = 0.25
Step 2. Multiply the vector magnitude by the scalar to get new the magnitude.
the magnitude of ` bb"V"_(bb"10mya") =150 ((mm)/(yr)) xx 0.25 = 38 ((mm)/(yr))`
Step 3. Does your answer make sense?
The rate of 38 mm/yr is indeed much slower than the original 150 mm/yr.
Fig. 3. Blue vector shows the motion of the GPS station called TABL in California. White line shows the San Andreas Fault.
Provenance: Sarah Kruse, University of South Florida
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Problem 3. GPS station TABL lies just 3 km from the San Andreas fault, but it is not moving parallel to the fault (Fig. 3). The TABL station vector has magnitude (rate) = 26 mm/yr and an azimuth of 316
o. The San Andreas fault nearby has an azimuth of 295
o. What is the component of the velocity at the TABL site parallel to the San Andreas fault? What is the component of the velocity at TABL perpendicular to the San Andreas fault?
Step 1. 1a. Identify the resultant vector and the direction for which you want to find the perpendicular and parallel components.
1b. Find the angle between the resultant vector and the given direction.
- Resultant vector magnitude = ` bb"V"_bb"T" = 26` mm/yr .
- Resultant vector azimuth = `alpha = 316`o.
- Given direction = local azimuth of San Andreas fault `= 295`o.
- Angle beta between resultant vector and San Andreas fault `= 316` o `- 295` o `= 21` o .
Step 2. Find the triangle that includes the resultant, components parallel and perpendicular to the given direction, and an angle inside the triangle. To do this, it can help to rotate your geometry to draw the given direction as vertical or horizontal
Fig. 4. Blue vector shows motion of the GPS station TABL. White line shows the San Andreas Fault.
Provenance: Sarah Kruse, University of South Florida
Reuse: This item is in the public domain and maybe reused freely without restriction.
In Figure 4 the image is rotated so the San Andreas fault is horizontal. This is just to help visualize the triangle so that the desired components are horizontal and vertical. The yellow triangle includes the resultant, components, and the angle beta.
Step 3. Use trigonometry to solve for the components.
The yellow triangle on Figure 4 includes the resultant
` bb"V"_(bb"T")` , components
` bb"V"_(bb"PERP")` and
` bb"V"_(bb"PAR")` , and an angle within the triangle.
- `sin(beta) = (opp)/(hyp) = bb"V"_(bb"PERP")/bb"V"_bb"T"`. Rearranging, `bb"V"_(bb"PERP") = sin(beta)*bb"V"_bb"T" = 0.36 * 26` mm/yr `= 9.4` mm/yr .
- `cos(beta) = (adj)/(hyp) = bb"V"_(bb"PAR")/bb"V"_bb"T"`. Rearranging, `bb"V"_(bb"PAR")= cos(beta)*bb"V"_bb"T" = 0.93 * 26` mm/yr `= 24` mm/yr .
Step 4. Does your answer make sense?
Check 1: Should the parallel and perpendicular components of motion have faster or slower rates than the resultant ` bb"V"_(bb"T")` ? Component magnitudes will always be smaller than the resultant magnitude, as can be seen from the yellow triangle. 9.4 mm/yr perpendicular and 24 mm/yr northward are indeed slower than the resultant 26 mm/yr. Check 2: Just looking at the direction of ` bb"V"_(bb"T")` , should the perpendicular component be greater (faster) than the parallel? No, the plate at the TABL site is moving more in a direction closer to parallel than perpendicular to the San Andreas fault. The perpendicular component of motion is only 9.4 mm/yr, while the parallel (24 mm/yr) is almost as great as the resultant motion of 26 mm/yr.
When will a sinkhole collapse?
Fig. 5. Famous sinkhole collapse in Winter Park, FL in 1981.
![[creative commons]](/images/creativecommons_16.png)
Provenance: https://www.chicagotribune.com/nation-world/os-fla360-pictures-winter-park-sinkhole-20121113-photogallery.html
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.
Fig. 6. Sinkhole force balance.
Provenance: Sarah Kruse, University of South Florida
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A sinkhole is a cavern in limestone. The roof will remain stable if the sum of the forces is upward, but if conditions change such that the sum of the forces becomes downward, the roof will collapse. The forces are shown in Figure 6 .
Problem 4: Use vector addition to sum the forces on the roof of the cavern for the case when the weight of the roof is 6.1 x 106 N, the cavern is filled with water which provides a buoyancy force of 1.2 x 106 N, and the cohesion on the sides of the roof is 5.2 x 106 N. Will this sinkhole collapse?
Step 1. Identify the magnitude and direction of your initial vectors.
We need a coordinate system to define the direction of vectors. This problem isn't 2D (like North and East for map vectors), but is just 1D (all forces are vertical). For the example we'll define vertical up as positive, so vectors in the opposite direction (down) will appear as negative. With this coordinate system, the vectors to be summed are
- ` bb"F"_bb"w" "weight of roof" = -6.1 xx 10^6 N "(down = negative up)" `
- ` bb"F"_bb"b" "buoyancy force" = 1.2 xx 10^6 N "(up)" `
- ` bb"F"_bb"c" "cohesion" = 5.2 xx 10^6 N "(up)" `
Step 2. Break your vectors into the respective direction components.
The only direction is up-down, and we defined these in Step 1.
Step 3. Add the magnitudes of the components with the same direction.
The resultant upward force
` bb"F"_(bb"sum")= bb"F"_bb"w" + bb"F"_bb"b" + bb"F"_bb"c" = -6.1 xx 10^6 N + 1.2 xx 10^6 N + 5.2 xx 10^6 N = 0.3 xx 10^6 N`
Step 4. We don't need to find the magnitude and azimuth as shown in step 4 of the example problem because this problem just has one direction (up-down). The magnitude of the resultant vector is just the sum in Step 3, and the direction is up because the resultant is positive. This sinkhole will not collapse.
Step 5. Does your answer make sense?
The resultant sum in Step 3 is positive upward. Just checking, the sum of the two upward forces supporting the roof is `1.2 xx 10^6 N + 5.2 xx 10^6 N = 6.4 xx 10^6 N` (pink plus blue vectors in Figure 6). This is greater than the downward force of 6.1 x 106 N (black vector in Figure 6), so greater upward force should keep the roof stable.
Relative tectonic plate motion
A special case of vectors arises in the Earth sciences when describing a velocity vector
relative to another velocity vector,. This requires a calculation that involves the velocities of two objects that are both in motion. So, the symbol for the velocity of A with respect to fixed B would be:
`\ _ (bb"B") bb"V"_(bb"A")` . The velocity of B with respect to fixed A would be:
`\ _ (bb"A") bb"V"_(bb"B")` .
To understand this notation imagine you are on a train and you see a car driving on an adjacent road. The velocity of the car relative to you on the moving train would be `\ _ (bb"Train") bb"V"_(bb"Car")` . To a passenger in the car, your relative velocity appears as `\ _ (bb"Car") bb"V"_(bb"Train")` . The velocity of the reference location is written as the first subscript.
The relative velocity vectors `\ _ (bb"B") bb"V"_(bb"A")` and `\ _ (bb"A") bb"V"_(bb"B")` are identical but point in opposite directions, so `\ _ (bb"B") bb"V"_(bb"A") = -\ _ (bb"A") bb"V"_(bb"B")` .
To accurately describe the motion of tectonic plates on Earth their motions need to be placed in a reference frame. Earth scientists often do this by describing the velocity of plates
relative to another plate. Here we will explore this concept and practice our vector math along the San Andreas fault in central California.
Problem 5: The San Andreas Fault is commonly thought of as the representation of the boundary between the North American and Pacific tectonic plates. Near Parkfield, CA, the GPS stations have recorded the Pacific Plate (Station 1) moving at a rate of 46 mm/yr in the direction of 323o (NW) and the North American Plate (Station 2) is moving 23 mm/yr in the same direction (Fig. 7). To find the relative velocity of a plate A with respect to a plate B, subtract the velocity of plate B from the velocity of plate A. (Vector subtraction is just like vector addition, but multiply the magnitude of the subtracting vector by -1.)
What is the relative velocity of the Pacific plate with respect to North America near Parkfield, CA? What is the relative velocity of the North American plate with respect to the Pacific?
Fig. 7. Map of Pacific and North American tectonic plates near Parkfield, CA.
Provenance: John Zayac, Vassar 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.
Step 1. Identify the magnitude and direction of each vector given in the problem and determine the vector needed to be solved for.
Here the vectors have the same direction so we can represent them in a simple map. The velocity vector for the Pacific plate
`bb"V"_(bb"P")` has a rate of 46 mm/yr with an azimuth of 323
o and the North American plate motion
`bb"V"_(bb"NA")` has the same azimuth moving at a rate of 23 mm/yr. We want to solve for:
- `\ _ (bb"NA") bb"V"_(bb"P")` = the velocity of the Pacific plate with respect to North America.
- `\ _ (bb"P") bb"V"_(bb"NA")` = the velocity of the North American plate with respect to the Pacific.
Step 2. Break down each of the given vectors into their respective directional components.
Vector map illustrating the relationships between the vectors in the problem.
Provenance: John Zayac, Vassar College
Reuse: This item is in the public domain and maybe reused freely without restriction.
Since the vectors share the same azimuth, this step is not necessary.
Add the magnitudes of the component vectors with the same direction to calculate the component vectors of the sum.
Vector map illustrating the relationships between the vectors in the problem.
Provenance: John Zayac, Vassar College
Reuse: This item is in the public domain and maybe reused freely without restriction.
Since the plates have the same azimuth in this location, we can perform scalar operations without having to break the vectors into their components.
So, we can perform the calculation as follows:
`\ _ (bb"NA") bb"V"_(bb"P") = (bb"V"_bb"P") - (bb"V"_(bb"NA")) = 46 `mm/yr` - 23 `mm/yr` = 23 `mm/yr with an azimuth of `323`o.
`\ _ (bb"P") bb"V"_(bb"NA")= (bb"V"_(bb"NA")) - (bb"V"_bb"P") = 23 `mm/yr` - 46 `mm/yr` = -23 `mm/yr with an azimuth of `323`o.
- `\ _ (bb"P") bb"V"_(bb"NA")` can also be expressed as 23 mm/yr with an azimuth of 143o.
Step 4. Use your results in Step 3 to find the magnitude and azimuth of the sum vector.
This step is not necessary because the vectors share the same azimuth.
Step 5. Does your answer make sense?
Here we see that the relative plate motions near Parkfield are equal in magnitude, but opposite in direction, as they should be. Also, since the Pacific plate is moving faster in the 323
o direction, relative to North America it should be moving in that same direction with a positive value.
The boundary between the Pacific and North American plates in California is a classic example of a transform plate boundary, where the plates are sliding past each other in opposite directions. Our answer is consistent with that understanding.
Motion of sand grains entrained by longshore transport
The transport of sand at the beach is controlled by wind-driven waves running up the beach face (the swash) and the water flowing back down the beach under the influence of gravity (the backwash). If the waves approach the shore perfectly perpendicular to the beach, the motion is simply one-dimensional. However, most waves approach the shore at an angle which causes motion that is parallel to the beach as well. We can examine the overall motion of the sand grains using the principles of vector math.
Provenance: https://www.internetgeography.net/topics/what-is-longshore-drift/
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.
Problem 6: What is the velocity of the backwash and the net eastward velocity of a sand grain being transported along a perfectly oriented east-west beach face if a wave strikes the beach with a swash velocity of 3 meters per second (m/s) with an azimuth of 26o? Assume that the grain has the same velocity as the water, there is only east-west net transport, and that the direction of backwash is perpendicular to the beach face.
Step 1. Identify the magnitude and direction of each vector given in the problem and determine the vector needed to be solved for.
The vector representing the velocity of the swash (
` bb"V"_(bb"S")` ) has a magnitude of 3 meters per second (m/s) and an azimuth of 26
o.
The vector representing the velocity of the backwash ( ` bb"V"_(bb"BW")` ) has a magnitude equal to the north component of ` bb"V"_(bb"S")` and an azimuth of 180o (which is the opposite direction). See the accompanying vector map to make sure this makes sense to you.
The net motion of the sand grain will be ` bb"V"_(bb"Net")` which will be the equivalent of the east component of ` bb"V"_(bb"S")` . This cuts down on the math we need to perform to solve the problem.
Step 2. Break down each of the given vectors into their respective directional components.
We can calculate the the north and east component vectors of
` bb"V"_(bb"S")` (denoted as
` bb"V"_(bb"S(N)")` and
` bb"V"_(bb"S(E)")` respectively) using trigonometry.
` bb"V"_(bb"S"(bb"N")) = bb"V"_bb"S"*cos(alpha) = 3 `m/s` * cos(26`o`) = 2.70 `m/s
` bb"V"_(bb"S"(bb"E")) = bb"V"_bb"S"*sin(alpha) = 3 `m/s` * sin(26`o`) = 1.32 `m/s
Step 3. Add the magnitudes of the component vectors with the same direction to calculate the component vectors of the sum.
We do not need to perform this step in this case.
Step 4. Use your results in Step 3 to find the magnitude and azimuth of the sum vector.
Using the relationships discovered in Step 1, we can now solve for both
` bb"V"_(bb"BW")` and
` bb"V"_(bb"Net")` using the component vectors of
` bb"V"_(bb"S")` we determined in Step 2.
` bb"V"_(bb"BW") = -(bb"V"_(bb"S"("N"))) = -(2.70 `m/s`) = 2.70 `m/s with an azimuth of `180`o
` bb"V"_(bb"net") = (bb"V"_(bb"S"(bb"E"))) = 1.32 `m/s with an azimuth of `90`o
Step 5. Does your answer make sense?
Our magnitudes for ` bb"V"_(bb"BW")` and ` bb"V"_(bb"Net")` should be less than ` bb"V"_(bb"S")` and this is true for our results. In addition, the net velocity should be parallel to the beach face (it is) and the backwash should be perpendicular (it is) - so our answer is consistent.
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