The parallelogram method of subtracting vectors is a graphical technique used to find the difference between two vectors. It relies on the principle that subtracting one vector from another is equivalent to adding the negative of the second vector to the first.
Understanding the Method
Vector subtraction, such as finding the difference between vector $\mathbf{a}$ and vector $\mathbf{b}$ ($\mathbf{a} - \mathbf{b}$), can be conceptually understood as adding vector $\mathbf{a}$ and the negative of vector $\mathbf{b}$ ($\mathbf{a} + (-\mathbf{b})$). The negative of a vector has the same magnitude (length) but the opposite direction.
The parallelogram method applies the standard parallelogram law of addition to this concept of adding the negative vector.
The Parallelogram Law of Subtraction
According to the provided reference, "if two vectors a and -b are starting from a point P and they are two adjacent sides of a parallelogram, then their sum which is a + (-b) (which can also be written as a - b) is the vector that represents the diagonal of a parallelogram that starts from P".
This definition highlights the core idea:
- Identify the two vectors involved, say $\mathbf{a}$ and $\mathbf{b}$.
- Determine the negative of the vector being subtracted, which is $-\mathbf{b}$.
- Apply the parallelogram law of addition to the two vectors $\mathbf{a}$ and $-\mathbf{b}$.
Steps to Subtract Vectors Using the Parallelogram Method
To graphically subtract vector $\mathbf{b}$ from vector $\mathbf{a}$ (i.e., find $\mathbf{a} - \mathbf{b}$) using this method, follow these steps:
- Choose a Starting Point (P): Draw both vector $\mathbf{a}$ and vector $-\mathbf{b}$ starting from the same point P. Remember that $-\mathbf{b}$ has the same length as $\mathbf{b}$ but points in the exact opposite direction.
- Form the Parallelogram: Use vector $\mathbf{a}$ and vector $-\mathbf{b}$ as two adjacent sides originating from point P. Complete the parallelogram by drawing lines parallel to $\mathbf{a}$ and $-\mathbf{b}$ from the endpoints of $-\mathbf{b}$ and $\mathbf{a}$ respectively.
- Draw the Resultant Vector: Draw the diagonal of the parallelogram that starts from the common origin point P.
- Identify the Difference Vector: This diagonal vector, starting from P, represents the resultant vector $\mathbf{a} + (-\mathbf{b})$, which is equal to $\mathbf{a} - \mathbf{b}$.
Visualizing the Process
Imagine you have vector $\mathbf{a}$ pointing northeast and vector $\mathbf{b}$ pointing east.
- First, you would find $-\mathbf{b}$, which points west.
- Then, you place the tails of $\mathbf{a}$ and $-\mathbf{b}$ at the same point.
- You complete the parallelogram using these two vectors as sides.
- The diagonal drawn from the shared tails to the opposite corner of the parallelogram gives you the vector $\mathbf{a} - \mathbf{b}$.
This method provides a clear graphical representation of vector subtraction by transforming it into a vector addition problem involving the negative vector.
