Since we know how to add vectors, subtracting them is straightforward. Subtracting a vector is the same as adding its opposite. Just as in scalar arithmetic where $5 - 3$ is the same as $5 + (-3)$, subtracting vector $\vec{B}$ from vector $\vec{A}$ is the same as adding $-\vec{B}$ to $\vec{A}$.

This simple rule is the key to both graphical and algebraic subtraction. Let's see how it works.

To subtract vectors graphically, we use the same tip-to-tail method we used for addition, but with a slight modification. First, you draw vector $\vec{A}$. Then, you find the opposite of vector $\vec{B}$ by drawing a vector of the same length but pointing 180 degrees in the other direction. Finally, you place the tail of $-\vec{B}$ at the tip of $\vec{A}$. The resultant vector, $\vec{R}$, goes from the tail of $\vec{A}$ to the tip of $-\vec{B}$.

Another way to think about this visually is to ask: what vector do I need to add to $\vec{B}$ to get $\vec{A}$? If you place the tails of $\vec{A}$ and $\vec{B}$ at the same origin, the vector that points from the tip of $\vec{B}$ to the tip of $\vec{A}$ is the resultant vector $\vec{A} - \vec{B}$. This is often called the triangle rule for subtraction.

This second method is particularly useful for finding a displacement vector. If an object moves from a point defined by position vector $\vec{r}_1$ to a point defined by position vector $\vec{r}_2$, the displacement vector $\Delta\vec{r}$ is $\vec{r}_2 - \vec{r}_1$.

Subtracting vectors by their components is even more direct. Just as with addition, you operate on the corresponding components. To find $\vec{A} - \vec{B}$, you subtract the components of $\vec{B}$ from the components of $\vec{A}$.

Let's use an example. Suppose a boat is moving with velocity $\vec{v}_{boat} = \langle 10, 5 \rangle$ meters per second relative to the water, and the river current has a velocity of $\vec{v}_{current} = \langle 2, -1 \rangle$ m/s.

What is the boat's velocity relative to the river bank? That's simple addition. But what if we want to know the velocity of the water relative to the boat? We would need to calculate $\vec{v}_{current} - \vec{v}_{boat}$.

From the perspective of someone on the boat, the water seems to be rushing backward at 8 m/s and sideways at 6 m/s. This calculation is crucial in fields like aviation and marine navigation to determine relative motion.

Let's check your understanding of these concepts.

Vector subtraction is a fundamental operation that builds on the principles of vector addition. By treating subtraction as the addition of an opposite vector, we can solve a wide range of problems involving changes in position, velocity, or force.