Method

The Parallelogram Method

The parallelogram method is a way to find the sum of two or more vectors. For example, consider the vectors shown below, whose component forms are

a = ⟨ - 1, 3 ⟩

and

b = ⟨ 2, 1 ⟩ .

The following three steps can be applied to find the sum of $a$ and $b .$

Reposition the Vectors

Translate

b

so that the two vectors have the same initial point.

Draw a Parallelogram

Draw two line segments, each parallel to one of the vectors, so that a parallelogram is obtained with

a

and

b

as adjacent sides.

This step is why this method is called the Parallelogram Method!

Draw the Diagonal

Consider the diagonal of the parallelogram from the common point of the two vectors. The vector that has its tail at this point and its tip at the opposite vertex of the parallelogram is the sum of the two vectors.

The sum of the two vectors is also a vector called the resultant.

Notice that the component form of the resultant is

⟨ 1, 4 ⟩

and it coincides with the vector addition of

a

and

b .

a + b = ⟨ - 1, 3 ⟩ + ⟨ 2, 1 ⟩ = ⟨ - 1 + 2, 3 + 1 ⟩ = ⟨ 1, 4 ⟩

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