Vectors in Two and Three Dimensions

Introduction

Vectors give you a compact way to describe quantities that have both magnitude and direction. In two dimensions, a vector can model a displacement across a map or a velocity on a flat surface. In three dimensions, vectors can describe the motion of an aircraft, the force on a structure, the direction of a ray in computer graphics, or the orientation of an object in space.

In this aiMOOC for Grades 11–13, you will move between geometric pictures and coordinate calculations. You will learn how to represent vectors, combine them, measure them, compare their directions, and use them to describe lines and planes. You will also meet the dot product and the cross product, two operations that connect algebra with geometry.

The diagram above shows vector addition geometrically. The same sum can be found by adding coordinates component by component.

As you watch, focus on the three viewpoints of a vector: an arrow in space, a list of coordinates, and an object that can be added or scaled.

Learning Goals

By the end of this course, you should be able to explain and apply the following ideas:

Vector: Interpret vectors geometrically and in coordinate form in two and three dimensions.
Vector addition: Add, subtract, and scale vectors and connect the calculations to geometric constructions.
Magnitude: Compute vector lengths and construct unit vectors.
Dot product: Use the dot product to test perpendicularity, determine angles, and calculate projections.
Cross product: Compute a three-dimensional vector perpendicular to two given vectors and interpret its magnitude as an area.
Line: Describe lines parametrically with a position vector and a direction vector.
Plane: Describe planes using points and normal vectors and solve line–plane intersection problems.
Mathematical modelling: Apply vectors to motion, forces, work, torque, geometry, and spatial reasoning.

Foundations: Points, Scalars, and Vectors

Scalars and Vectors

A scalar is described by magnitude alone. Examples include mass, temperature, time, and energy. A vector has magnitude and direction. Examples include displacement, velocity, acceleration, and force.

A vector is often written as $\vec{a}$ . In two-dimensional Cartesian coordinates, $\vec{a} = (a_{1}, a_{2})$ . In three dimensions, $\vec{a} = (a_{1}, a_{2}, a_{3})$ .

Coordinates tell you how much of the vector points along each coordinate axis. Two vectors are equal when their corresponding components are equal, even if the arrows are drawn at different positions.

Position Vectors and Displacement Vectors

A point and a vector are related but are not the same object. The point $P = (x, y, z)$ identifies a location. Its position vector is the vector from the origin to that point: $\vec{O P} = (x, y, z)$ .

If $A = (a_{1}, a_{2}, a_{3})$ and $B = (b_{1}, b_{2}, b_{3})$ , then the displacement from $A$ to $B$ is $\vec{A B} = B - A = (b_{1} - a_{1}, b_{2} - a_{2}, b_{3} - a_{3})$ .

This subtraction rule is one of the most useful ideas in analytic geometry: final position minus initial position gives the displacement vector.

Vector Operations in Two Dimensions

Addition and Subtraction

For $\vec{a} = (a_{1}, a_{2})$ and $\vec{b} = (b_{1}, b_{2})$ , $\vec{a} + \vec{b} = (a_{1} + b_{1}, a_{2} + b_{2})$ and $\vec{a} - \vec{b} = (a_{1} - b_{1}, a_{2} - b_{2})$ .

Geometrically, addition can be carried out by the head-to-tail rule or the parallelogram rule. Subtraction can be understood as addition of the opposite vector: $\vec{a} - \vec{b} = \vec{a} + (- \vec{b})$ .

When you read a vector diagram, distinguish carefully between the coordinates of a point and the components of an arrow.

Scalar Multiplication

For a scalar $k$ and a vector $\vec{a}$ , $k \vec{a} = (k a_{1}, k a_{2})$ in two dimensions and $k \vec{a} = (k a_{1}, k a_{2}, k a_{3})$ in three dimensions.

If $k > 1$ , the vector becomes longer. If $0 < k < 1$ , it becomes shorter. If $k < 0$ , the direction reverses as well as the magnitude changing. Scalar multiplication therefore stretches, shrinks, or reverses a vector while keeping it parallel to the original.

Components, Magnitude, and Unit Vectors

For $\vec{a} = (a_{1}, a_{2})$ , the magnitude follows from the Pythagorean theorem: $| \vec{a} | = \sqrt{a_{1}^{2} + a_{2}^{2}}$ .

For $\vec{a} = (a_{1}, a_{2}, a_{3})$ , $| \vec{a} | = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$ .

If $\vec{a} \neq \vec{0}$ , the unit vector in the direction of $\vec{a}$ is $\hat{a} = \frac{\vec{a}}{| \vec{a} |}$ . A unit vector has magnitude 1.

The component picture is especially useful when a vector is given by a magnitude and an angle. In two dimensions, a vector of magnitude $r$ making an angle $θ$ with the positive x-axis has components $(r \cos θ, r \sin θ)$ .

Worked Example: A Displacement in Two Dimensions

Let $A = (- 1, 2)$ and $B = (3, 5)$ . Then $\vec{A B} = (3 - (- 1), 5 - 2) = (4, 3)$ .

Its magnitude is $| \vec{A B} | = \sqrt{4^{2} + 3^{2}} = 5$ .

A unit vector from $A$ toward $B$ is $\frac{\vec{A B}}{| \vec{A B} |} = (\frac{4}{5}, \frac{3}{5})$ .

Notice the chain of reasoning: identify two points, subtract to find displacement, use the Pythagorean theorem for magnitude, then divide by the magnitude to normalize the vector.

Vectors in Three Dimensions

The Three-Dimensional Coordinate System

Three-dimensional Cartesian space uses three mutually perpendicular axes, usually called x, y, and z. A point $P = (x, y, z)$ is located by three coordinates, and a vector $\vec{v} = (v_{x}, v_{y}, v_{z})$ records its components along the same axes.

Fehler beim Erstellen des Vorschaubildes:

A drawing of three-dimensional axes on a flat page uses perspective, so lengths and angles in the picture may not look exact. The coordinates, not the appearance of the sketch, determine the mathematics.

Fehler beim Erstellen des Vorschaubildes:

This decomposition shows how one spatial vector can be resolved into components parallel to the coordinate axes.

Distance and Direction in Space

For points $P = (x_{1}, y_{1}, z_{1})$ and $Q = (x_{2}, y_{2}, z_{2})$ , the displacement vector is $\vec{P Q} = (x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$ .

The distance between the points is the magnitude of this displacement: $P Q = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}$ .

The same vector operations used in two dimensions extend naturally to three dimensions because components are still added, subtracted, and scaled one coordinate at a time.

The Dot Product

Algebraic and Geometric Definitions

For $\vec{a} = (a_{1}, a_{2}, a_{3})$ and $\vec{b} = (b_{1}, b_{2}, b_{3})$ , the dot product is $\vec{a} \cdot \vec{b} = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ .

Geometrically, $\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos θ$ , where $θ$ is the angle between the two nonzero vectors.

Therefore, $\cos θ = \frac{\vec{a} \cdot \vec{b}}{| \vec{a} | | \vec{b} |}$ .

If the dot product of two nonzero vectors is positive, the angle between them is acute. If it is zero, the vectors are perpendicular. If it is negative, the angle is obtuse.

Datei:Scalar-product-dot-product.svg

The diagram connects the dot product to projection: one vector contributes according to how much of it lies in the direction of the other.

While watching, compare the coordinate formula with the geometric idea of projection.

Projection

The vector projection of $\vec{a}$ onto a nonzero vector $\vec{b}$ is ${proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b}$ .

This result is parallel to $\vec{b}$ . Projection is useful for splitting a vector into a component parallel to a chosen direction and a component perpendicular to it.

Datei:Scalar vector projection.svg

In physics, work done by a constant force $\vec{F}$ over a displacement $\vec{s}$ is $W = \vec{F} \cdot \vec{s}$ . Only the component of force in the direction of motion contributes to this dot product.

Worked Example: Angle Between Two Spatial Vectors

Let $\vec{a} = (2, - 1, 3)$ and $\vec{b} = (1, 4, - 2)$ . Their dot product is $\vec{a} \cdot \vec{b} = 2 - 4 - 6 = - 8$ .

Their magnitudes are $| \vec{a} | = \sqrt{14}$ and $| \vec{b} | = \sqrt{21}$ . Hence $\cos θ = \frac{- 8}{\sqrt{14} \sqrt{21}} = \frac{- 8}{\sqrt{294}}$ .

Because the cosine is negative, the angle is obtuse. This conclusion can be made even before calculating the angle numerically.

The Cross Product in Three Dimensions

Definition and Direction

For three-dimensional vectors $\vec{a} = (a_{1}, a_{2}, a_{3})$ and $\vec{b} = (b_{1}, b_{2}, b_{3})$ , the cross product is $\vec{a} \times \vec{b} = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})$ .

The vector $\vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ . Its direction follows the right-hand rule.

Datei:Cross-product-right-hand rule.svg

Its magnitude is $| \vec{a} \times \vec{b} | = | \vec{a} | | \vec{b} | \sin θ$ . This equals the area of the parallelogram spanned by the two vectors. Half of that magnitude gives the area of the triangle determined by the same two side vectors.

The cross product is not commutative: $\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})$ .

Use the video to connect the coordinate formula, perpendicular direction, and oriented area.

This second explanation gives practice calculating both dot and cross products with unit-vector notation.

Worked Example: A Perpendicular Vector

Using $\vec{a} = (2, - 1, 3)$ and $\vec{b} = (1, 4, - 2)$ , we obtain $\vec{a} \times \vec{b} = (- 10, 7, 9)$ .

Check perpendicularity: $(- 10, 7, 9) \cdot (2, - 1, 3) = - 20 - 7 + 27 = 0$ , and $(- 10, 7, 9) \cdot (1, 4, - 2) = - 10 + 28 - 18 = 0$ .

Both dot products are zero, confirming that the cross product is perpendicular to each original vector.

Lines in Two and Three Dimensions

Vector Equation of a Line

A line through a point with position vector $\vec{p}$ and nonzero direction vector $\vec{d}$ can be written $\vec{r} = \vec{p} + t \vec{d}$ , where $t$ is a real parameter.

In three dimensions, if $\vec{p} = (p_{1}, p_{2}, p_{3})$ and $\vec{d} = (d_{1}, d_{2}, d_{3})$ , then $(x, y, z) = (p_{1}, p_{2}, p_{3}) + t (d_{1}, d_{2}, d_{3})$ .

This gives the parametric equations $x = p_{1} + t d_{1}$ , $y = p_{2} + t d_{2}$ , and $z = p_{3} + t d_{3}$ .

Two lines with parallel direction vectors are parallel or coincident. In three dimensions, nonparallel lines may intersect, but they may also be skew: they pass through different parts of space without meeting.

Line Segments and Midpoints

A line segment from point $A$ to point $B$ can be parametrized by $\vec{r} (t) = A + t (B - A)$ for $0 \leq t \leq 1$ .

At $t = 0$ , the point is $A$ . At $t = 1$ , it is $B$ . At $t = \frac{1}{2}$ , the point is the midpoint: $M = \frac{A + B}{2}$ .

This form is widely used in animation, computer graphics, and numerical modelling because varying the parameter smoothly traces the segment.

Planes in Three Dimensions

Normal-Vector Form

A plane is determined by a point $P_{0}$ and a nonzero normal vector $\vec{n}$ perpendicular to the plane. A point with position vector $\vec{r}$ lies on the plane exactly when $\vec{n} \cdot (\vec{r} - {\vec{r}}_{0}) = 0$ .

If $\vec{n} = (a, b, c)$ , this can be rearranged to $a x + b y + c z = d$ for a suitable constant $d$ .

Two planes with parallel normal vectors are parallel or identical. If their normals are not parallel, the planes intersect in a line.

Finding a Plane from Three Points

Suppose three noncollinear points are $A$ , $B$ , and $C$ . Form two direction vectors in the plane: $\vec{A B} = B - A$ and $\vec{A C} = C - A$ .

Then $\vec{n} = \vec{A B} \times \vec{A C}$ is perpendicular to the plane. Use this as the normal vector in $\vec{n} \cdot (\vec{r} - A) = 0$ .

This method shows how subtraction, cross products, and dot products work together in three-dimensional geometry.

Worked Example: Line–Plane Intersection

Consider the line $\vec{r} = (1, 2, 0) + t (2, - 1, 3)$ and the plane $x + 2 y - z = 8$ .

Substitute the line coordinates into the plane: $(1 + 2 t) + 2 (2 - t) - 3 t = 8$ .

After simplification, $5 - 3 t = 8$ , so $t = - 1$ .

The intersection point is therefore $(1, 2, 0) - (2, - 1, 3) = (- 1, 3, - 3)$ . Checking gives $- 1 + 2 \cdot 3 - (- 3) = 8$ , so the point lies on the plane.

Areas, Volumes, and Orientation

Area from a Cross Product

If adjacent sides of a parallelogram are represented by $\vec{a}$ and $\vec{b}$ , its area is $| \vec{a} \times \vec{b} |$ .

If those vectors form two sides of a triangle from the same vertex, the triangle area is $\frac{1}{2} | \vec{a} \times \vec{b} |$ .

In two dimensions, a closely related determinant $a_{1} b_{2} - a_{2} b_{1}$ gives a signed area and encodes orientation.

Volume from a Scalar Triple Product

For three three-dimensional vectors $\vec{a}$ , $\vec{b}$ , and $\vec{c}$ , the scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ has absolute value equal to the volume of the parallelepiped spanned by the vectors.

If the scalar triple product is zero, the three vectors are coplanar. This gives an efficient algebraic test for whether three directions lie in the same plane.

Applications and Mathematical Modelling

Motion and Navigation

A position vector can vary with time: $\vec{r} (t)$ . In calculus, its derivative is the velocity vector and the derivative of velocity is acceleration. Even before calculus, displacement vectors can be added to model successive stages of a journey.

In navigation, you must distinguish between a vehicle's velocity relative to the surrounding medium and the velocity of the medium itself. Adding vectors can model the combined effect of a boat and current or an aircraft and wind.

Forces, Work, and Torque

Forces combine by vector addition. If several forces act on an object, their vector sum is the resultant force.

The dot product models work: $W = \vec{F} \cdot \vec{s}$ .

The cross product models torque: $\vec{τ} = \vec{r} \times \vec{F}$ , where $\vec{r}$ points from the pivot to the point where the force is applied. The torque direction is perpendicular to the plane containing $\vec{r}$ and $\vec{F}$ .

Computer Graphics and Spatial Technology

Three-dimensional graphics use position vectors, direction vectors, normal vectors, and transformations. Surface normals help determine how light interacts with a surface. Rays can be described by parametric line equations. Interpolation along segments creates smooth motion between positions.

Robotics and engineering use vectors to describe positions, orientations, velocities, and forces. A good vector model always begins by defining a coordinate system and interpreting the meaning and units of each component.

Problem-Solving Strategy

A Reliable Workflow

When solving a vector problem, you can use this sequence:

Coordinate system: Define axes, positive directions, units, and the meaning of each coordinate.
Diagram: Sketch the geometry and label points, vectors, angles, and known lengths.
Representation: Convert the geometric information into points, components, or vector equations.
Operation: Choose addition, subtraction, scalar multiplication, dot product, cross product, or a parametric equation according to the goal.
Verification: Check dimensions, signs, magnitudes, perpendicularity, and whether your result satisfies the original conditions.

Common Errors to Avoid

Confusing points with vectors: a point is a location, while a vector represents a displacement or direction.

Reversing a displacement: $\vec{A B} = B - A$ , whereas $\vec{B A} = A - B = - \vec{A B}$ .

Forgetting to normalize: a direction vector is not automatically a unit vector.

Using the dot and cross products interchangeably: the dot product returns a scalar, while the three-dimensional cross product returns a vector.

Ignoring order in a cross product: changing the order reverses the direction.

Trusting a perspective sketch too much: a 3D drawing on paper may distort lengths and angles.

Dividing by a zero magnitude: the zero vector has no defined direction and cannot be normalized.

Interactive Tasks

Quiz: Test Your Knowledge

Which statement best describes a vector? (A quantity with magnitude and direction) (!A quantity with magnitude only) (!A point with no coordinates) (!A number that must be positive)

How do you find the displacement vector from point A to point B? (Subtract the coordinates of A from the coordinates of B) (!Add all coordinates of A and B) (!Subtract the coordinates of B from the coordinates of A) (!Multiply corresponding coordinates)

What does dividing a nonzero vector by its magnitude produce? (A unit vector in the same direction) (!The zero vector) (!A perpendicular vector) (!A scalar equal to one)

When is the dot product of two nonzero vectors equal to zero? (When the vectors are perpendicular) (!When the vectors are parallel) (!When the vectors have equal magnitude) (!When the vectors have opposite components)

What kind of result does a dot product produce? (A scalar) (!A point) (!A plane) (!A three dimensional axis)

What is true about the direction of a three dimensional cross product? (It is perpendicular to both input vectors) (!It is always parallel to the first vector) (!It is always parallel to the second vector) (!It always points toward the origin)

What does the magnitude of a cross product represent geometrically? (The area of the parallelogram spanned by the vectors) (!The perimeter of the parallelogram) (!The angle between the coordinate axes) (!The distance from the origin to a plane)

What does a direction vector specify in a parametric line equation? (The direction in which the line extends) (!The unique midpoint of the line) (!The normal direction of every plane) (!The area enclosed by the line)

Which vector is most useful for describing the orientation of a plane? (A normal vector) (!A zero vector) (!A midpoint vector) (!A scalar vector)

What does a zero scalar triple product show? (The three vectors are coplanar) (!The three vectors all have unit length) (!The three vectors are mutually perpendicular) (!The three vectors point in the same direction)

Memory Game

Vector magnitude	Length of a vector
Unit vector	Vector with magnitude one
Dot product	Scalar measuring directional alignment
Cross product	Perpendicular vector determined by two spatial vectors
Position vector	Vector from the origin to a point
Direction vector	Vector parallel to a line

Drag and Drop

Match the correct terms.	Topic
Component addition	Combine vectors coordinate by coordinate
Normalization	Scale a nonzero vector to unit length
Orthogonality test	Use a zero dot product for nonzero vectors
Area calculation	Use the magnitude of a cross product
Line parametrization	Combine a fixed point with a scaled direction vector

Match each operation with the description that best explains its role in vector geometry.

Crossword Puzzle

Vector	What mathematical object has both magnitude and direction?
Scalar	What type of result is produced by a dot product?
Orthogonal	Which word describes two perpendicular directions?
Projection	What operation extracts the component of one vector along another?
Parallelogram	Which shape has area equal to the magnitude of a cross product of adjacent side vectors?
Direction	What does a nonzero vector specify for a parametric line?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Vector Photo Hunt: Find four real situations around your school or home that can be represented by vectors, photograph or sketch them, and label a possible magnitude, direction, and unit for each.
Coordinate Arrow Poster: Create a one-page poster showing a two-dimensional vector, its x- and y-components, its magnitude, and its unit vector.
Human Vector Walk: Mark a coordinate grid on the floor or outdoors, follow a sequence of displacement vectors, and record how vector addition predicts your final location.
Vector Vocabulary Explanation: Write a short explanation for a younger learner that distinguishes a point, scalar, vector, magnitude, and direction using one coherent example.

Standard

Wind and Motion Model: Build a numerical model of an aircraft or boat affected by wind or current, calculate the resultant velocity, and explain the meaning of each component.
Dot Product Investigation: Choose several vector pairs, calculate their dot products and angles, and create a diagram or spreadsheet showing how the sign of the dot product changes with angle.
Three-Dimensional String Model: Construct a physical model of three coordinate axes with string or rods, represent at least three spatial vectors, and explain their components from two viewing angles.
Line Intersection Video: Produce a short tutorial video showing how to decide whether two given lines in two or three dimensions intersect, are parallel, or are skew.

Advanced

Plane Reconstruction Project: Choose three noncollinear points in space, derive the plane through them using a cross product, and verify the equation with all three points.
Torque Experiment: Design a safe lever experiment with measured distances and forces, model the torque vector conceptually, and discuss how changing the angle changes torque magnitude.
Graphics Normal Vector Study: Investigate how surface normal vectors are used in three-dimensional computer graphics, create a simple visual demonstration, and explain the role of dot products in lighting.
Vector Modelling Investigation: Formulate a realistic multi-step problem involving motion, lines, or planes, solve it using vector methods, validate the result, and present the assumptions and limitations of your model.

Learning Assessment

Vector Representation Assessment: Given a geometric situation, define a coordinate system, represent all relevant points and vectors, and justify why your representation is appropriate.
Strategy Selection Assessment: For several problems, decide whether addition, a dot product, a cross product, or a parametric equation is the best tool, and explain each choice before calculating.
Reasoning with Perpendicularity: Use both dot-product calculations and geometric arguments to test claims about perpendicular vectors, then explain why the methods agree.
Line and Plane Transfer Task: Solve a line–plane intersection problem in three dimensions, verify the point algebraically, and interpret the parameter geometrically.
Modelling Critique: Analyse a vector model of wind, force, or motion, identify its assumptions, and propose one realistic refinement.
Error Analysis: Diagnose incorrect solutions involving displacement order, normalization, or cross-product order, correct them, and explain the mathematical consequence of each error.
Synthesis Challenge: Use subtraction, a cross product, and a dot product in one connected solution to determine a plane from points and test whether another point lies on it.

Evidence of Learning

Strong evidence of learning includes:

Knowledge: You can explain vectors, components, magnitude, direction, unit vectors, dot products, cross products, parametric lines, and normal-vector planes in your own words.
Skills: You can move accurately between diagrams, coordinates, equations, and geometric interpretations in two and three dimensions.
Reasoning: You can justify why a chosen vector operation fits a problem and verify results with independent checks such as substitution, magnitude, or perpendicularity.
Products: You can produce clear diagrams, models, written solutions, investigations, videos, or digital visualizations that communicate vector relationships.
Transfer: You can apply vector methods to unfamiliar contexts in physics, engineering, navigation, computer graphics, or spatial geometry.
Reflection: You can identify limitations in a model, explain errors, and improve a solution after checking assumptions and units.

OERs on the Topic

The following embedded English Wikipedia article provides an open reference for the concept of Euclidean vectors and links to related ideas in geometry and linear algebra.

Use this reference to review terminology, compare notations, and follow links to related topics such as Dot product, Cross product, Vector space, and Analytic geometry.