Resultant Vector, Sum of Vectors

How to calculate the resultant vector

The resultant vector is the vector that 'results' from adding two or more vectors together. There are a two different ways to calculate the resultant vector.

Methods for calculating a Resultant Vector
picture of resultant vector

In the picture on the left, the black vector is the resultant of the two red vectors. To try to understand what a resultant is consider the following story.

If you drove from your house, centered at the origin. To your friends house, at the point (3,4), imagine that you had to take two different roads these are the two red vectors. However, the resultant vector vector would be the straight line path from your home to your friend's house, and the black vector represents that path.

Head to Tail Method

The head to tail method is way to find the resultant vector. The steps are quite straight forward. The head to tail method considers the head of a vector to be the end with the arrow, or the 'pointy end'. The tail of the vector is where the vector begins.
  • Place the two vectors next to each other such that the head of the one vector is touching the tail of the other vector.
  • Draw the resultant vector by starting where

Steps for Head to Tail Method

head to tail method animated gif

Calculate the magnitude resultant vector

Find the sum of each pair of vectors (the magnitude of the resultant vector).

picture of resultant vector

To find the resultant vector's magnitude, use the pythagorean theorem

Calculate resultant vector

Practice Problems

Problem 1

You left your house to visit a friend. You got in your car drove 40 miles east, then got on a highway and went 50 miles north.

Draw a vector from the beginning of your journey, your home, and the end, your friend’s house.

Resultant Vector Real world problem

How long is the vector that you drew?

resultant answer
Problem 2

What is the sum of the two vectors? Use the head to tail method to calculate the resultant vector in the picture on the right

Head to Tail Practice

Parallelogram Method to Calculate Resultant

Before tackling the parallelogram method for solving resultant vectors, you should be comfortable with the following topics

Problem 3

To best understand how the parallelogram method works, lets examine the two vectors below. The vectors have magnitudes of 17 and 28 and the angle between them is 66°. Our goal is to use the parallelogram method to determine the magnitude of the resultant.

Parallelogram method
Step 1

Step 1) Draw a parallelogram based on the two vectors that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them)

Step 2

Step 2) We now have a parallelogram and know two angles (opposite angles of parallelograms are congruent). We can also figure out the other pair of angles since the other pair are congruent and all four angles must add up to 360.

Step 3

Step 3 Draw the paprallelograms diagonal. This diagonal is the resultant vector

Step 4

Use the law of cosines to determine the length of the resultant

Use the law of cosines to calculate the resultant.