# How do I use vectors?Plotting magnitude and direction in the Earth sciences

## What are vectors and what are they good for?

To describe the temperature, it's enough to say it's 65 degrees F. If all that is required is a number and units, we call that thing a scalar quantity. But to describe something that is moving, or a force causing motion, the direction also matters. Groundwater is flowing at a rate of 1.5 feet per day (ft/d) 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; scalar quantities are in regular font.

## How do I distinguish vectors from scalars and understand vector maps?

Specially mounted GPS instruments use satellite data to measure the local velocity of the tectonic plate they are on. In this example you will use the map to the right, showing GPS velocity vectors, to identify whether plates are rigid or are internally deforming, and identify active plate boundaries.

Vectors have both a magnitude and a direction

• Magnitude is the number, in this case the rate of motion of the GPS station in millimeters per year (mm/yr)
• Direction of the vector is the orientation of the arrow
• On vector maps , usually the tail of the arrow marks the point that the vector describes.

In this module we define map directions using azimuth.

Maps Step 1. This map shows elevations and plate motion velocities. Is each property a scalar or a vector?

Maps Step 2. Identify the magnitude scale of vectors. What is the rate of motion bb "V"_bb"C" of the GPS on the Cocos Islands (the longest arrow in the lower right)?

Maps Step 3. 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?

Bold text, such as  bb"M"  , is used here to show a vector quantity.  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 directions.

## How do I find the components of a vector?

Finding vector components requires trigonometry.

• You can review Trigonometry.
• Make sure you know whether your calculator expects angles in degrees or radians.  Get help Using your calculator.
• Note: Excel in default mode uses angles in radians. 1  "radian" = pi/180 "degrees" . In Excel pi is written as pi().

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.

The Cocos Islands station velocity vector has magnitude (rate) = 116 mm/yr and an azimuth of 36o. What is the eastward rate of motion of the Cocos Island GPS station? What is the northward rate of motion?

Components Step 1 (maps). Identify the resultant vector magnitude and direction, and the components you want to solve for.

Components Step 2 (maps). Find the triangle that includes the resultant, components, and angle.

Components Step 3 (maps). Use trigonometry to solve for the components.

### To find vector components relative to another direction

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.

Intuitively, we would guess that the steeper the dip of the rupture 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 on the landslide bb"G" and the dip angle of the slip 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") acts to drive slip of the body of the landslide down the rupture surface. In contrast the perpendicular bb"G"_(bb"p") helps the landslide stick to the underlying rock, so it acts to resist 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.

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. 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.

Components 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 may help to rotate your geometry to draw the given direction as vertical or horizontal.)

Components Step 3. Use the trigonometry in Figure AB to solve for the components.

## 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 .

### × Velocity of a Volcanic Bomb

Explosive volcanic eruptions are dynamic events where material is ejected from the volcanic conduit at high velocities. Volcanologists refer to the material erupted as tephra, which is classified based on its size. Large (> 2 mm) projectiles are called 'bombs' and in high energy eruptions can fall kilometers (km) away from the vent.

A volcanic bomb is erupted from a volcano with a 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. What is the velocity of the bomb just before impact?

Step 1. Identify the magnitude and direction of the vector to be multiplied and the scalar multiplication factor.

Step 2. Multiply the vector magnitude by the scalar to get new the magnitude.

## How do I add vectors?

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.

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 prior to 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 the 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?

Step 1. Identify the magnitude and direction of each vector given in the problem and determine the vector needed to be solved for.

Step 2. Break down each of the given vectors into their respective directional components.

Step 3. Add the magnitudes of the component vectors with the same direction to calculate the component vectors of the sum.

Step 4. Use your results in Step 3 to find the magnitude and azimuth of the sum vector.

## 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

If you think you have a handle 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).