Table of Contents
In the parent chapter on vectors in the plane, you will already have met what a vector is and how to represent one (for example as an arrow in the plane or as a pair of numbers like $(a,b)$). Here we focus specifically on adding vectors.
Geometric idea of vector addition
Vector addition combines two (or more) vectors to produce a resultant vector.
Suppose you have vectors $\vec{u}$ and $\vec{v}$ drawn as arrows in the plane. To add them geometrically:
- Place the tail of $\vec{v}$ at the head of $\vec{u}$ (this is called the “tip-to-tail” method).
- Draw a new vector from the tail of $\vec{u}$ to the head of $\vec{v}$.
- This new vector is $\vec{u} + \vec{v}$.
It does not matter which order you do this in:
- Put $\vec{v}$ first and then attach $\vec{u}$ at its head, or
- Put $\vec{u}$ first and then attach $\vec{v}$.
In both constructions the resultant arrow (from the first tail to the last head) is the same. This shows visually that
$$
\vec{u} + \vec{v} = \vec{v} + \vec{u}.
$$
You can also see vector addition as the diagonal of a parallelogram:
- Draw $\vec{u}$ and $\vec{v}$ starting from the same point.
- Complete the parallelogram with $\vec{u}$ and $\vec{v}$ as adjacent sides.
- The diagonal from the common starting point to the opposite corner is $\vec{u} + \vec{v}$.
Both the tip‑to‑tail and parallelogram constructions describe the same operation.
Coordinate rule for adding vectors in the plane
In the plane, a vector is often represented by an ordered pair of real numbers:
$$
\vec{u} = (u_1, u_2), \quad \vec{v} = (v_1, v_2).
$$
Vector addition is defined componentwise:
$$
\vec{u} + \vec{v} = (u_1 + v_1,\; u_2 + v_2).
$$
You add the $x$‑components to get the new $x$‑component, and the $y$‑components to get the new $y$‑component.
If you want to add more than two vectors,
$$
\vec{u} = (u_1,u_2), \quad \vec{v} = (v_1,v_2), \quad \vec{w} = (w_1,w_2),
$$
then
$$
\vec{u} + \vec{v} + \vec{w} = (u_1 + v_1 + w_1,\; u_2 + v_2 + w_2).
$$
This is consistent with doing the additions in steps, for example
$$
(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}),
$$
because addition is associative (see below).
Properties of vector addition
Vector addition in the plane has several key algebraic properties. These mirror familiar properties of real numbers, but here they apply to vectors.
Let $\vec{u}, \vec{v}, \vec{w}$ be vectors in the plane.
- Commutativity
The order of addition does not matter:
$$
\vec{u} + \vec{v} = \vec{v} + \vec{u}.
$$
Geometrically, the parallelogram picture or switching the order of the tip‑to‑tail construction gives the same resultant vector. - Associativity
Grouping does not matter:
$$
(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}).
$$
This means you can add several vectors in any order and grouping and get the same final vector. In coordinates, both sides give
$$
(u_1 + v_1 + w_1,\; u_2 + v_2 + w_2).
$$ - Additive identity (zero vector)
There is a special vector, the zero vector, usually written $\vec{0} = (0,0)$, such that
$$
\vec{u} + \vec{0} = \vec{0} + \vec{u} = \vec{u}.
$$
Geometrically, adding $\vec{0}$ does nothing: the resultant is the same arrow you started with. - Additive inverse (negative of a vector)
For each vector $\vec{u}$ there is an opposite vector $-\vec{u}$ such that
$$
\vec{u} + (-\vec{u}) = \vec{0}.
$$
In coordinates, if $\vec{u} = (u_1,u_2)$, then
$$
-\vec{u} = (-u_1, -u_2).
$$
Geometrically, $-\vec{u}$ is the same length as $\vec{u}$ but points in the exact opposite direction.
These properties are fundamental: they allow you to manipulate expressions involving sums of vectors confidently, rearranging and regrouping them as needed.
Relation between vector addition and subtraction
Subtraction of vectors is defined using addition and the additive inverse:
$$
\vec{u} - \vec{v} = \vec{u} + (-\vec{v}).
$$
In coordinates, if $\vec{u} = (u_1,u_2)$ and $\vec{v} = (v_1,v_2)$, then
$$
\vec{u} - \vec{v} = (u_1 - v_1,\; u_2 - v_2).
$$
Geometrically, you can think of $\vec{u} - \vec{v}$ as the vector you add to $\vec{v}$ to get $\vec{u}$:
$$
\vec{v} + (\vec{u} - \vec{v}) = \vec{u}.
$$
So if you draw $\vec{u}$ and $\vec{v}$ from the same tail, then $\vec{u} - \vec{v}$ is the arrow from the head of $\vec{v}$ to the head of $\vec{u}$.
Using vector addition to combine displacements and forces
Vector addition is especially useful for quantities that have both magnitude and direction, such as displacements and forces.
- Displacements: If you walk according to displacement $\vec{a}$, then displacement $\vec{b}$, your overall displacement from where you started is $\vec{a} + \vec{b}$.
- Forces: If two forces $\vec{F}_1$ and $\vec{F}_2$ act on an object, the overall or resultant force is
$$
\vec{F}_{\text{resultant}} = \vec{F}_1 + \vec{F}_2.
$$
This is again found by tip‑to‑tail addition or by adding components.
In each case, the same mathematical operation—vector addition—describes how separate directional effects combine into one.
Summary
In the plane, vector addition:
- Is defined geometrically by tip‑to‑tail placement or the parallelogram construction.
- Is defined algebraically by adding corresponding components.
- Satisfies commutativity, associativity, and has a zero vector and additive inverses.
- Naturally describes combining displacements, velocities, forces, and other directional quantities.
These ideas underlie more advanced operations with vectors, including those treated in the chapter on the dot product.