Vector Subtraction

We've explored vector addition and its key properties. Now, leveraging the close relationship between vector addition and scalar addition (and the connection between addition and subtraction), we can define vector subtraction using a combination of vector addition and negative vectors.

The core idea is that subtracting a number is the same as adding its negative. We'll apply this same principle to vectors.

Rethinking Subtraction as Addition

Consider the scalar subtraction $9 - 4$ . We can rewrite this as the addition of 9 and the negative of 4:

9 - 4 \= 9 + (-4)

Recall from the previous article that $-4$ is the additive inverse of $4$ .

Defining Vector Subtraction

We apply this same idea to vectors $\overrightarrow{X}$ and $\overrightarrow{Y}$ . We define vector subtraction as:

\overrightarrow{X} - \overrightarrow{Y} \= \overrightarrow{X} + (-\overrightarrow{Y})

In words: Subtracting vector $\overrightarrow{Y}$ from vector $\overrightarrow{X}$ is the same as adding the negative of vector $\overrightarrow{Y}$ to vector $\overrightarrow{X}$ .

Visualizing Vector Subtraction

To visualize this, let's consider two position vectors $\overrightarrow{X}$ and $\overrightarrow{Y}$ starting at the origin $O$ .

To perform the subtraction $\overrightarrow{X} - \overrightarrow{Y}$ :

Find the negative of $\overrightarrow{Y}$ ( $-\overrightarrow{Y}$ ): This is a vector with the same magnitude as $\overrightarrow{Y}$ but pointing in the opposite direction.
Add $\overrightarrow{X}$ and $-\overrightarrow{Y}$ using the head-to-tail method: Place the tail of $-\overrightarrow{Y}$ at the head of $\overrightarrow{X}$ . The resultant vector, which goes from the tail of $\overrightarrow{X}$ to the head of $-\overrightarrow{Y}$ , is the vector $\overrightarrow{X} - \overrightarrow{Y}$ .

Alternative Visual Interpretation

There is an alternative way to visualise vector subtraction. The vector $\overrightarrow{X} - \overrightarrow{Y}$ can also be viewed as the vector that points from the head of $\overrightarrow{Y}$ to the head of $\overrightarrow{X}$ , when these vectors have been placed tail-to-tail.

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