Vector Subtraction
Subtracting two vectors is a simple operation, and it works in a way that's very similar to vector addition.
To calculate the difference between two vectors, just subtract the corresponding components of each vector.
For example, if you have two vectors \( \vec{v} \) and \( \vec{w} \):
$$ \vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix} $$
$$ \vec{w} = \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{pmatrix} $$
The subtraction \( \vec{v} - \vec{w} \) is defined as a new vector where each component is the difference of the corresponding components:
$$ \vec{v} - \vec{w} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix} - \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{pmatrix} = \begin{pmatrix} v_1 - w_1 \\ v_2 - w_2 \\ \vdots \\ v_n - w_n \end{pmatrix} $$
Example: Consider two vectors:
$$ \vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \quad \vec{w} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} $$
The difference vector is obtained by subtracting the components of \( \vec{w} \) from \( \vec{v} \):
$$ \vec{v} - \vec{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 - 1 \\ 4 - 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix} $$
Another way to think of subtraction is as the addition of \( \vec{v} \) with the opposite vector of \( \vec{w} \).
Let’s break it down with a practical example:
If you have the following two vectors:
$$ \vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \quad \vec{w} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} $$
You can rewrite the subtraction as adding \( \vec{v} \) to the negative of \( \vec{w} \):
$$ \vec{v} - \vec{w} = \vec{v} + (-\vec{w}) $$
$$ \vec{v} - \vec{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} + \left(-\begin{pmatrix} 1 \\ 2 \end{pmatrix}\right) $$
$$ \vec{v} - \vec{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} + \begin{pmatrix} -1 \\ -2 \end{pmatrix} $$
$$ \vec{v} - \vec{w} = \begin{pmatrix} 3 - 1 \\ 4 - 2 \end{pmatrix} $$
$$ \vec{v} - \vec{w} = \begin{pmatrix} 2 \\ 2 \end{pmatrix} $$
From a geometric perspective, subtracting two vectors in a plane involves drawing a vector from the tip of \( \vec{w} \) to the tip of \( \vec{v} \), with \( \vec{v} \) anchored at the origin.
For instance, plot the vectors \( \vec{v} = (3, 4) \) and \( \vec{w} = (1, 2) \) from the example above:
To visualize \( \vec{v} - \vec{w} \), draw a vector connecting \( \vec{w} \) to \( \vec{v} \). Then, translate this vector so it starts at the origin (O).
The result is the difference vector.
Alternatively, you can calculate \( \vec{v} - \vec{w} \) using the head-to-tail method. This involves adding \( \vec{v} \) to the opposite vector \( -\vec{w} \).
No matter the method, the result is always the same.
I hope this explanation helps! If you have more questions about vectors, feel free to ask.
