Vectors As Arrows In The Plane

Walking from the origin to the point $(3,2)$ and walking from the origin to $(1,-1)$ are two different displacements. Each displacement has a direction and a length, and it does not depend on any coordinate grid. A vector in the plane is this geometric displacement, drawn as an oriented arrow. For most of this course, we will draw vectors based at the origin so that we can compare them and combine them without worrying about where they started.

Two operations drive almost everything that comes later. You can place one arrow head to tail with another and get a new arrow, and you can stretch or reverse an arrow by multiplying by a number. These two moves are the geometric roots of matrix multiplication in $\mathbb{R}^2$, and later we will name the transformations that preserve them.

Try dragging and stretching the arrows and watch what stays invariant under sliding an arrow without rotating it.

Coordinate pairs as names for arrows

Once arrows all start at the origin, a coordinate pair becomes a convenient label. The arrow that ends at $(3,2)$ is a different object from the ordered pair $(3,2)$, but they correspond perfectly when a coordinate system is fixed. The grid provides two special arrows, one unit to the right and one unit up. Call them the basis vectors $e_1$ and $e_2$. Concretely, $e_1$ is the arrow to $(1,0)$ and $e_2$ is the arrow to $(0,1)$.

Now the arrow to $(3,2)$ can be described by how many copies of $e_1$ and $e_2$ you need. Three steps of $e_1$ move you $3$ units in the $x$ direction, and two steps of $e_2$ move you $2$ units in the $y$ direction. That observation motivates writing the vector as

$$3e_1+2e_2.$$

The numbers $3$ and $2$ are the components of the vector in this basis, and the ordered pair $(3,2)$ is the coordinate name for the same arrow. This is a dictionary, not a definition of a vector. Change the basis, and the same arrow gets a different pair of components.

A common mistake is to treat $(3,2)$ as the vector itself, as if vectors are just pairs of numbers. The pair depends on the coordinate frame, while the arrow does not. The course will use coordinates constantly, but the geometry is the object that is invariant.

Adding and scaling arrows

Head to tail addition answers a concrete question. If one displacement takes you from the origin to $v$, and a second displacement takes you from where you are to a new location, what single displacement takes you from the origin to the final point? Place the second arrow so its tail starts at the first arrow’s head. The arrow from the origin to the new head is the sum $v+w$.

When both arrows start at the origin, the same construction becomes the parallelogram rule. Draw the parallelogram with sides $v$ and $w$. Its diagonal from the origin is $v+w$. The key geometric fact is that addition is independent of how you slide arrows around, as long as you keep their direction and length.

Scalar multiplication answers a different question. If you keep the direction fixed, how do you scale the length? Multiplying a vector $v$ by a scalar $c$ stretches it by a factor of $|c|$. If $c<0$, it also reverses direction. So $2v$ doubles the length, $\tfrac12 v$ halves it, and $-v$ flips it.

Play with addition and scaling and look for the moment when $cv$ switches direction and when $v+w$ lines up with one of the inputs.