Vectors - Practice Problems

Solving Earth science problems with direction and magnitude

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.

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?

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?

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?

Movement of tectonic plates

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 14^o.  What are the magnitude and direction of the velocity vector ` bb"V"_(bb"10mya")` for motion 10 million years ago?

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?

When will a sinkhole collapse?

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?  

Relative tectonic plate motion

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  323^o (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?

 

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.

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 26^o? 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.

 

Next steps

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