Vector Subtraction

Subtracting a Vector by Adding Its Opposite

Vector subtraction gives the difference between two vectors. When we subtract $\vec{b}$ from $\vec{a}$, we obtain the vector that must be added to $\vec{b}$ to produce $\vec{a}$.

Mathematically, vector subtraction is defined as:

In other words, subtracting $\vec{b}$ from $\vec{a}$ is equivalent to adding $\vec{a}$ to the negative of $\vec{b}$.

Geometric Vector Subtraction

The following four steps show two equivalent constructions of $\vec{a} - \vec{b}$:

1. Draw $\vec{a}$ and $\vec{b}$ with the same initial point.
2. Draw the arrow from the endpoint of $\vec{b}$ to the endpoint of $\vec{a}$. That arrow is $\vec{a} - \vec{b}$.
3. For the second construction, reverse $\vec{b}$ to form $-\vec{b}$, then translate it so its initial point meets the endpoint of $\vec{a}$.
4. Draw the arrow from the common origin to the final endpoint. It is $\vec{a}+(-\vec{b})=\vec{a}-\vec{b}$.

The visualization shows both constructions. The two arrows labeled $a-b$ have the same components, length, and direction even though they begin at different points.

Algebraic Vector Subtraction

Algebraically, subtract corresponding components. Suppose we have two vectors:

Then vector subtraction $\vec{a} - \vec{b}$ can be calculated as:

For two-dimensional vectors, the equation becomes:

Example of Vector Subtraction Calculation

Suppose there are two vectors:

Subtract $\vec{a} - \vec{b}$ component by component:

While vector subtraction $\vec{b} - \vec{a}$ is:

Notice that $\vec{a} - \vec{b} \neq \vec{b} - \vec{a}$ in general. Reversing the order reverses the direction: $\vec{a} - \vec{b} = -(\vec{b} - \vec{a})$.

Applications of Vector Subtraction

Vector subtraction appears whenever we compare two vector quantities:

1. Calculating Displacement: If $\vec{a}$ is the final position and $\vec{b}$ is the initial position, then $\vec{a} - \vec{b}$ is the displacement vector.

2. Calculating Distance: In a game, subtracting the bird's position from the target's position gives the direction toward the target. The magnitude of that difference vector gives the straight-line distance.

3. Comparing Velocities: In physics, the velocity of object $A$ relative to object $B$ is $\vec{v}_{A/B}=\vec{v}_A-\vec{v}_B$. Reversing the order reverses the relative-velocity vector.

Vector Subtraction to Find Displacement

The displacement between two positions is found by subtracting the initial position vector from the final position vector.

Suppose an object moves from initial position $\vec{p}_1$ to final position $\vec{p}_2$. The displacement vector of the object is:

This vector points from the initial position to the final position. Its magnitude is the straight-line distance between those positions, which may be shorter than the path actually traveled.

Properties of Vector Subtraction

The rule $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$ turns a subtraction into an addition, so you can reuse every rule you already know for adding vectors. Subtracting a vector from itself always gives the zero vector. Reversing the two vectors reverses the sign of the result, so keep the order of the two vectors in mind.

1. $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$
2. $\vec{a} - \vec{a} = \vec{0}$ (zero vector)
3. $\vec{a} - \vec{0} = \vec{a}$
4. $\vec{0} - \vec{a} = -\vec{a}$
5. $\vec{a} - \vec{b} = -(\vec{b} - \vec{a})$

Finding a Displacement by Vector Subtraction

A monkey is at position $(3, 4)$ and wants to get a banana located at position $(2, 1)$. Determine the displacement vector of the monkey to reach the banana.

Solution:

Monkey's position: $\vec{m} = (3, 4)$

Banana's position: $\vec{p} = (2, 1)$

Displacement vector of the monkey to the banana:

The monkey needs to move $1 \text{ unit}$ in the negative $x$-axis direction and $3 \text{ units}$ in the negative $y$-axis direction to reach the banana.

The magnitude of the displacement vector can be calculated using the Pythagorean theorem: