Resultant Vector: Definition & Formula

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  • 0:07 What is a Resultant Vector?
  • 1:11 A Simple Example
  • 2:00 Calculating Example 1
  • 3:43 Calculating Example 2
  • 5:22 Lesson Summary
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Lesson Transcript
Instructor: Emily Cadic

Emily has a master's degree in engineering and currently teaches middle and high school science.

In this lesson, you'll learn about resultant vectors and when they should be used. You'll also find out how to work with the head-to-tail method and have the chance to apply your new knowledge to some practice problems.

What Is a Resultant Vector?

A resultant vector is the combination of two or more single vectors. When used alone, the term vector refers to a graphical representation of the magnitude and direction of a physical entity like force, velocity, or acceleration.

If the definition of a vector alone does not jog your memory, think about the single process of opening a door. First, you have to exert enough force to actually move the door, but that's only part of the story, the magnitude part. You also have to figure out which direction to move the door, in other words, whether to push or pull.

While magnitude is expressed as a numerical value, direction can be expressed in a variety of ways: both qualitative—like north, south, left and right—or quantitative with a system or coordinates or angles. The purpose of a resultant vector is to report solutions in the most concise manner possible. It may appear in your math studies or in physical science problems dealing with forces and motion.

A Simple Example

Let's explore a very simple example to help you get comfortable with resultant vectors. Take a look at the forces being applied in the following picture:

If all you knew was that each of the porters was pushing with a force of 50 pounds, would you be able to figure out the weight of the master plus the vehicle? You could if you were using resultant vectors!

The resultant vector is the total of the four individual vectors, or 200 pounds, which should be equal and opposite to the weight of the master and the vehicle. The resultant vector of parallel vectors can be found by adding the vectors if they are pointing in the same direction, as seen here, or by subtracting the vectors if they are pointing opposite one another.

Calculating Example 1

Our first example did not require any special formula because it dealt with parallel vectors. Now, let's see how we can calculate the resultant vector of two orthogonal, or perpendicular, vectors using trigonometry.

For example, on your way to work in the morning, you drive five miles to a gas station (Vector A) before heading another ten miles north to your office (Vector B). Assuming you travel directly from work at the end of the day, what is the resultant vector of your drive home?

This problem can be solved in five steps:

Step 1: Line up the head of the first vector with the tail of the second vector, which is automatic for this problem.

Step 2: Connect the two vectors with a line that becomes the hypotenuse of a right triangle.

Step 3: Find the length of the hypotenuse using the Pythagorean theorem; this is the magnitude of the resultant vector.

Step 4: Find the angle between the tail of the hypotenuse and the adjacent vector; this is the direction of the resultant vector.

Do you remember the mnemonic device, SOH-CAH-TOA, you used in trigonometry? You can use any of the three methods to calculate the angle, but TOA is a good option because the opposite and adjacent sides of the triangle are both nice whole numbers.

Step 5: Report your final answer. The resultant vector of your drive home has a magnitude of 11.2 miles and a direction of 26.6 degrees southwest.

Calculating Example 2

Sometimes, you may encounter a resultant vector problem that is less straightforward than the previous example, such as angled vectors. Before you start to panic, just break down the vector with the decomposition method shown in the next example:

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