Vector Addition

Forces, velocities, and accelerations can all be represented as vectors and you will often need to find the sum of these vectors during problem solving. To do this you will need to add the vectors.

The sum of any number of vectors can be determined geometrically using the following strategy. Starting with one of the vectors as the base, we redraw the second vector so that the tail of the second vector begins at the tip of the first vector. We can repeat this with a third vector, putting the tail of the third vector at the tip of the second vector. From here we repeat this pattern until all vectors are drawn this way. Once the vectors are all drawn out tip to tail, the sum of all the vectors will be the vector connecting the tail of the first vector to the tip of the last vector.

Geometric Vector Addition
The geometric addition of vectors involves putting the vectors tip to tail as shown above.

The easiest way to determine the magnitude and direction of the sum of the vectors is to add the vectors in component form. This means that we will need to separate each vector into x, y, and possibly z components. As we can see in the diagram below, the x component of the sum of all the vectors will be the sum of all the x components of the individual vectors. Similarly the y and z components of the sum of the vectors will be the sum of all the y components and the sum of all the z components respectively.

Vector Addition via Components
By summing all the components in a given direction, we can find the component of the sum of the vectors in that direction.

Once we find the sum of the sum of the vectors in component form we can convert this sum vector back to a magnitude and direction if necessary.

Worked Problems:

Question 1:

Determine the sum of the force vectors in the diagram below. Leave the sum in component form.

Problem 1 Diagram

Solution:



Question 2:

Determine the sum of the force vectors in the diagram below. Give the sum in terms of a magnitude and a direction.

Problem 2 Diagram

Solution:



Question 3:

Determine the sum of the force vectors in the diagram below. Leave the sum in component form.

Problem 3 Diagram

Solution: