Table of Contents
Vectors in the plane are a way to describe quantities that have both size (magnitude) and direction, using the familiar two–dimensional $xy$-coordinate system. In this chapter, we focus on how to represent, interpret, and use vectors in $\mathbb{R}^2$ (the Euclidean plane).
Geometric and Coordinate Representations
A vector in the plane can be pictured as an arrow:
- It has a tail (starting point) and a head (ending point).
- Its length represents its magnitude.
- The arrow points in the direction of the vector.
A key idea is that a vector does not depend on its exact location, only on its direction and length. Two arrows in the plane represent the same vector if you can slide one onto the other without rotating or stretching it.
In coordinates, a vector in the plane is written as an ordered pair
$$
\vec{v} = \langle v_1, v_2 \rangle
$$
or sometimes as
$$
\vec{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}.
$$
If a vector goes from point $A(x_1, y_1)$ to point $B(x_2, y_2)$, the corresponding vector $\overrightarrow{AB}$ has components
$$
\overrightarrow{AB} = \langle x_2 - x_1,\; y_2 - y_1 \rangle.
$$
The zero vector is the vector with no length and no direction:
$$
\vec{0} = \langle 0, 0 \rangle.
$$
Although it has no direction geometrically, it behaves like a neutral element in vector operations.
Basis Vectors and Component Form
In the standard Cartesian plane, we use the $x$- and $y$-axes. The standard basis vectors are
- $\vec{i} = \langle 1, 0 \rangle$ (one unit in the $x$-direction),
- $\vec{j} = \langle 0, 1 \rangle$ (one unit in the $y$-direction).
Any vector $\vec{v} = \langle v_1, v_2 \rangle$ can be written in terms of these as
$$
\vec{v} = v_1 \vec{i} + v_2 \vec{j}.
$$
This is just a different way of expressing the same information as the ordered pair.
Vector Addition and Scalar Multiplication (Geometric View)
Although the operations themselves will be studied in more generality elsewhere, it is useful here to see how they look in the plane.
Vector Addition
To add two vectors $\vec{u}$ and $\vec{v}$ geometrically, you can use either:
- Tip-to-tail method: Place the tail of $\vec{v}$ at the head of $\vec{u}$. The sum $\vec{u} + \vec{v}$ is the vector from the tail of $\vec{u}$ to the head of $\vec{v}$.
- Parallelogram method: Place the tails of $\vec{u}$ and $\vec{v}$ at the same point. Complete the parallelogram with $\vec{u}$ and $\vec{v}$ as adjacent sides. The diagonal from the common tail to the opposite corner is $\vec{u} + \vec{v}$.
In coordinates, addition is componentwise:
$$
\langle u_1, u_2 \rangle + \langle v_1, v_2 \rangle = \langle u_1 + v_1,\; u_2 + v_2 \rangle.
$$
Scalar Multiplication
A scalar is an ordinary real number. Multiplying a vector $\vec{v}$ by a scalar $k$ stretches or shrinks its length and possibly reverses its direction:
- If $k > 0$, $k\vec{v}$ has the same direction as $\vec{v}$ but is $|k|$ times as long.
- If $k < 0$, $k\vec{v}$ points in the opposite direction to $\vec{v}$ and is $|k|$ times as long.
- If $k = 0$, $k\vec{v} = \vec{0}$.
In coordinates,
$$
k \langle v_1, v_2 \rangle = \langle kv_1,\; kv_2 \rangle.
$$
Two vectors that are scalar multiples of each other are parallel (or anti-parallel if the scalar is negative).
Magnitude (Length) of a Vector
The magnitude or length of a vector $\vec{v} = \langle v_1, v_2 \rangle$ is denoted $|\vec{v}|$ or $\|\vec{v}\|$ and is given by
$$
|\vec{v}| = \sqrt{v_1^2 + v_2^2}.
$$
This comes from the Pythagorean theorem: if you draw the vector from the origin to $(v_1, v_2)$, it forms the hypotenuse of a right triangle with legs of lengths $|v_1|$ and $|v_2|$.
Basic properties:
- $|\vec{v}| \ge 0$ for all vectors $\vec{v}$.
- $|\vec{v}| = 0$ if and only if $\vec{v} = \langle 0, 0 \rangle$.
- For any scalar $k$, $|k\vec{v}| = |k|\;|\vec{v}|$.
Direction and Unit Vectors
The direction of a nonzero vector in the plane can be described using an angle from the positive $x$-axis, or by using a unit vector pointing in the same direction.
A unit vector is a vector of length $1$. If $\vec{v}$ is a nonzero vector, the unit vector in the direction of $\vec{v}$ is
$$
\hat{v} = \frac{\vec{v}}{|\vec{v}|}.
$$
This operation is often called normalizing the vector.
Example: If $\vec{v} = \langle 3, 4 \rangle$, then $|\vec{v}| = 5$ and the unit vector in its direction is
$$
\hat{v} = \left\langle \frac{3}{5},\; \frac{4}{5} \right\rangle.
$$
A unit vector in the direction making an angle $\theta$ with the positive $x$-axis has components
$$
\langle \cos\theta,\; \sin\theta \rangle.
$$
Dot Product in the Plane
The dot product (or scalar product) of two vectors in the plane is a way of combining them to get a scalar (a number), not another vector.
For vectors $\vec{u} = \langle u_1, u_2 \rangle$ and $\vec{v} = \langle v_1, v_2 \rangle$, the dot product is defined by
$$
\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2.
$$
This can also be related to the angle between the vectors:
If $\theta$ is the angle between nonzero vectors $\vec{u}$ and $\vec{v}$, then
$$
\vec{u} \cdot \vec{v} = |\vec{u}|\,|\vec{v}| \cos\theta.
$$
From this relationship:
- If $\vec{u} \cdot \vec{v} > 0$, the angle between them is acute ($0^\circ < \theta < 90^\circ$).
- If $\vec{u} \cdot \vec{v} = 0$, they are perpendicular (orthogonal), with $\theta = 90^\circ$.
- If $\vec{u} \cdot \vec{v} < 0$, the angle between them is obtuse ($90^\circ < \theta < 180^\circ$).
You can solve for the angle:
$$
\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}|\,|\vec{v}|}.
$$
Projections in the Plane
The dot product allows you to find how much one vector points in the direction of another. This is captured by the idea of a projection.
Let $\vec{u}$ and $\vec{v}$ be vectors with $\vec{v} \ne \vec{0}$.
- The scalar component of $\vec{u}$ in the direction of $\vec{v}$ is
$$
\operatorname{comp}_{\vec{v}}(\vec{u}) = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|}.
$$
This is a signed length: it can be positive or negative depending on whether $\vec{u}$ points generally in the same or opposite direction as $\vec{v}$. - The vector projection of $\vec{u}$ onto $\vec{v}$ is
$$
\operatorname{proj}_{\vec{v}}(\vec{u}) = \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2} \right) \vec{v}.
$$
Geometrically, $\operatorname{proj}_{\vec{v}}(\vec{u})$ is the vector you get by dropping a perpendicular from the head of $\vec{u}$ to the line through the origin in the direction of $\vec{v}$; it lies on that line.
Perpendicular and Parallel Vectors
In the plane, vector relationships often describe geometric configurations such as parallel and perpendicular lines.
- Two nonzero vectors $\vec{u}$ and $\vec{v}$ are parallel if there exists a scalar $k$ such that $\vec{u} = k\vec{v}$.
- They are perpendicular if $\vec{u} \cdot \vec{v} = 0$.
These conditions are especially useful in analytic geometry to:
- Determine if lines are parallel or perpendicular from direction vectors.
- Check orthogonality of sides of polygons (e.g., to verify right angles in rectangles and squares).
Using Vectors to Describe Lines in the Plane
Vectors offer a compact way to describe lines in the plane.
Let $\vec{r}$ denote a position vector of a point $(x, y)$, meaning
$$
\vec{r} = \langle x, y \rangle.
$$
Let $\vec{r}_0$ be the position vector of a fixed point $(x_0, y_0)$ on a line, and let $\vec{d}$ be a nonzero vector parallel to the line (a direction vector). Then every point on the line can be written as
$$
\vec{r} = \vec{r}_0 + t \vec{d}, \quad \text{for all real } t.
$$
This is called a vector equation of the line.
Writing this in component form:
If $\vec{r}_0 = \langle x_0, y_0 \rangle$ and $\vec{d} = \langle a, b \rangle$, then
$$
\langle x, y \rangle = \langle x_0, y_0 \rangle + t \langle a, b \rangle,
$$
which means
$$
x = x_0 + at, \quad y = y_0 + bt.
$$
These are parametric equations of the line, where $t$ is a real parameter.
Normal Vectors and Line Equations
A normal vector to a line is a vector perpendicular to the line. If $\vec{n} = \langle A, B \rangle$ is normal to a line and $(x_0, y_0)$ is a point on the line, then every point $(x, y)$ on the line satisfies
$$
\vec{n} \cdot \langle x - x_0, y - y_0 \rangle = 0.
$$
In coordinates, this becomes
$$
A(x - x_0) + B(y - y_0) = 0,
$$
which simplifies to the familiar equation of a line
$$
Ax + By + C = 0
$$
for some constant $C$.
Thus, vector ideas give both direction and normal forms of line equations:
- Direction vector $\vec{d}$: describes which way the line points.
- Normal vector $\vec{n}$: describes which way is perpendicular to the line.
Distance and Shortest Paths Using Vectors
Vectors also help describe distances and shortest paths between geometric objects in the plane.
Distance Between Two Points
If $A(x_1, y_1)$ and $B(x_2, y_2)$ are two points, the vector from $A$ to $B$ is
$$
\overrightarrow{AB} = \langle x_2 - x_1,\; y_2 - y_1 \rangle.
$$
The distance between $A$ and $B$ is just the magnitude of this vector:
$$
\text{distance}(A, B) = |\overrightarrow{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
$$
Shortest Distance from a Point to a Line (Idea)
Consider a point $P$ and a line $L$. To find the shortest distance from $P$ to $L$:
- Choose a point $Q$ on $L$.
- Use a normal vector $\vec{n}$ to $L$.
- Look at the vector $\overrightarrow{QP}$ and project it onto $\vec{n}$.
The magnitude of the component of $\overrightarrow{QP}$ in the direction of $\vec{n}$ gives the perpendicular distance from $P$ to $L. $
Although the detailed formulas belong elsewhere, this shows how vector projections connect directly to geometric distances.
Applications of Vectors in the Plane
Vectors in the plane appear naturally in many contexts:
- Physics: forces, velocities, accelerations, and displacements are all planar vectors in many problems.
- Navigation: representing movements in terms of east–west and north–south components.
- Computer graphics: describing motion, directions, and transformations of objects in the 2D plane.
- Analytic geometry: expressing lines, segments, and directions more flexibly than with scalar equations alone.
Throughout analytic geometry, vectors offer a powerful language to unify geometric and algebraic viewpoints in two dimensions, providing a foundation that generalizes naturally to higher dimensions.