The parallelogram method of subtracting vectors is a graphical technique that leverages the principle that subtracting a vector is equivalent to adding its negative counterpart. This allows you to use the familiar parallelogram method for addition to find the difference between two vectors.
According to the provided reference, for vectors $\vec{v}$ and $\vec{w}$, finding the difference $\vec{v} - \vec{w}$ is based on the fact that the negative of a vector has the opposite direction of the original vector and that adding a negative is equivalent to subtraction. Therefore, the core concept is represented by the equation:
v → − w → = v → + ( − w → )
This means that to subtract vector $\vec{w}$ from vector $\vec{v}$, you first find the negative of $\vec{w}$ (denoted as $-\vec{w}$) and then add $\vec{v}$ and $-\vec{w}$ using the standard parallelogram method for vector addition.
Key Concepts for Vector Subtraction via Parallelogram Method
Understanding this method relies on two fundamental vector concepts:
- Negative of a Vector: The negative of a vector $\vec{w}$, denoted as $-\vec{w}$, is a vector that has the same magnitude as $\vec{w}$ but points in the exact opposite direction.
- Subtraction as Addition: Vector subtraction $\vec{v} - \vec{w}$ is defined as the addition of vector $\vec{v}$ and the negative of vector $\vec{w}$, i.e., $\vec{v} + (-\vec{w})$.
Steps to Subtract Vectors Using the Parallelogram Method
To graphically subtract vector $\vec{w}$ from vector $\vec{v}$ using this method ($\vec{v} - \vec{w}$):
- Identify the Vectors: Start with the two vectors, $\vec{v}$ and $\vec{w}$.
- Find the Negative Vector: Determine the negative of the vector being subtracted, which is $-\vec{w}$. This vector will be parallel to $\vec{w}$, have the same length, but point in the reverse direction.
- Place Vectors at Origin: Draw both vector $\vec{v}$ and vector $-\vec{w}$ so that their initial points (tails) coincide at the same origin point.
- Construct the Parallelogram: Draw lines parallel to $\vec{v}$ and $-\vec{w}$ from the terminal points (heads) of $-\vec{w}$ and $\vec{v}$, respectively. These lines intersect to form a parallelogram.
- Draw the Resultant Vector: The diagonal of the parallelogram that starts from the common origin and extends to the opposite vertex represents the sum of $\vec{v}$ and $-\vec{w}$. This diagonal is the resultant vector, which is $\vec{v} - \vec{w}$.
This method provides a visual way to understand and calculate the difference between two vectors by converting the subtraction problem into an addition problem involving the negative vector.
