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Vector Addition in Physics

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Vector Addition

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Graphically Adding Vectors

When adding vectors in two dimensions, you set up the head or tip of the arrow to the tail of the other vector:

Graph of two vectors, representing two displacements of a person who walked 90 mi east and then 50 mi north

License: CC BY-NC 3.0

The sum of the two vectors is the vector that begins at the tail of the first vector and goes to the ending of the second vector:

Two vectors representing two displacements of a person who walked 90 mi east and then 50 mi north, with the third vector connecting the two, creating a triangle indicating the sum

License: CC BY-NC 3.0

You can find the length of the sum vector by using the Pythagorean theorem, a2 + b2 = c2. What would the sum, or resultant, vector be using the image above?

The direction can be found using a compass or by mathematical means.

How would you add three vectors graphically?

Mathematical Methods of Vector Addition

We can add vectors mathematically using trig, the law of cosines, or the Pythagorean theorem!

Remember: you can only use the Pythagorean theorem when vectors are at right angles to each other!

For vectors that are not at right angles to each other, we can resolve these vectors (blue and purple) into components (red) that lay on the axes:

Sum of two vectors on a north-south, east-west grid, creating a triangle for each vector

License: CC BY-NC 3.0

You now can find the magnitudes of the north-south and east-west component vectors using trig and find the respective sum components. What is the sum of the two east-west components based on the graph above? Of the two north-south components?

How can you use the Pythagorean theorem to find the resultant vector? What trig function would you use to find the direction of the resultant vector?

Remember: perpendicular vectors (i.e. north and east) have no components in the other direction and do not affect each other.

Click here to check your answers and find more information on vector addition in physics.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

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