Arrows and components, magnitude and direction, scalar multiplication, unit vectors, and the bridge from coordinate geometry to the language of linear algebra
This chapter introduces vectors — one of the foundational ideas of linear algebra and one of the most versatile mathematical tools across science, engineering, computer graphics, data science, and applied mathematics. Students who have studied coordinate geometry and trigonometry already have most of the background they need. This chapter makes that background precise, extends it into three dimensions, and connects geometric intuition to algebraic notation.
Vectors are not just arrows on a grid. They are ordered lists of numbers that can represent position, displacement, force, velocity, color, a data record, a digital signal, or any quantity that requires both magnitude and direction. Learning to move fluently between the geometric and algebraic views of a vector is one of the most important skills in linear algebra.
A ship leaves port and travels 40 miles east, then 30 miles north. How far is it from port, and in what direction? Now suppose a second ship leaves port and travels in a straight line to the same destination. What single displacement would describe that journey?
Questions like these — combining multiple movements, forces, or directions into a single description — are exactly what vectors are designed to answer. By the end of this chapter, you will be able to describe both journeys algebraically, compute the ship's distance from port, find the direction of travel, and extend the same ideas to three-dimensional space.
Algebra and precalculus built a powerful framework for working with numbers and one-variable functions. Linear algebra begins the transition to multiple quantities acting simultaneously. A vector is the simplest example of this transition: instead of a single number, you work with an ordered pair or triple of numbers that carry more information together than either number carries alone.
Vectors appear throughout mathematics and its applications. In physics, force, velocity, acceleration, and electric field are all vector quantities. In computer graphics, every point on a screen, every direction of light, and every transformation of a shape is described using vectors. In data science and machine learning, every data record is represented as a vector in a high-dimensional space. In navigation, GPS coordinates and travel directions are vector descriptions.
Understanding vectors in two and three dimensions is also essential preparation for later chapters in this book. Matrix multiplication, linear transformations, vector spaces, span, independence, projections, and eigenvectors all build on the vector foundations established here.
In arithmetic and algebra, the numbers you work with are scalars — single quantities described completely by their size. The number 5 is a scalar. A temperature of −3 degrees is a scalar. A speed of 60 miles per hour is a scalar. Scalars answer only the question "how much?"
Many real situations require more information than a single number can provide. A wind has both speed and direction. A force on a bridge has both strength and the angle at which it acts. A traveler's displacement has both distance and heading. A data record contains many measured quantities at once. These are situations that demand vectors.
A vector is a mathematical object that has both magnitude (size or length) and direction. The magnitude answers "how much?" and the direction answers "which way?" Together, these two pieces of information give a complete description.
Vectors are written in boldface: v, u, w. When writing by hand, use an arrow over the letter: v⃗. The magnitude of vector v is written ‖v‖ and is always a non-negative scalar.
| Type | Description | Example |
|---|---|---|
| Scalar | A single number with size only | Speed: 60 mph; Temperature: −3° |
| Vector | A quantity with both magnitude and direction | Velocity: 60 mph northeast; Force: 10 N upward |
Adding two scalars is straightforward: 5 + 3 = 8. Adding two vectors requires accounting for direction, not just magnitude. A force of 5 N to the right combined with a force of 3 N to the left is not 8 N — it is 2 N to the right. The direction changed the result completely. This is why vectors require their own arithmetic.
The most natural geometric picture of a vector is a directed arrow in the coordinate plane. A vector is drawn as an arrow from one point (the initial point or tail) to another point (the terminal point or tip). The length of the arrow represents the magnitude. The direction the arrow points represents the direction.
The arrow below starts at the origin P = (0, 0) and ends at Q = (3, 4). Its length is the distance from P to Q. Its direction is toward the upper right, at an angle determined by the components 3 and 4.
Vectors are not tied to a specific location. A fundamental property of vectors is that two arrows represent the same vector if they have the same length and point in the same direction — even if they start at different points. This is called free vector equivalence. The arrow from (0, 0) to (3, 4) represents the same vector as the arrow from (1, 1) to (4, 5), because both arrows have the same length and direction.
The special vector with magnitude zero and no defined direction is called the zero vector, written 0 (bold zero). It represents no displacement, no force, no movement.
Geometric arrows are useful for visualization, but algebra requires precise numerical descriptions. A vector in the coordinate plane is fully described by its components — the amounts it moves in the horizontal and vertical directions.
The vector from the origin (0, 0) to the point (a, b) is written as a column vector:
v = ( a )
( b )
where a is the x-component and b is the y-component. Given initial point P = (x₁, y₁) and terminal point Q = (x₂, y₂), the vector is v = (x₂ − x₁, y₂ − y₁) — the change in each coordinate.
In linear algebra, vectors are almost always written as column vectors with components stacked vertically. A row vector writes components horizontally. These behave differently in matrix calculations — a critical distinction that later chapters on matrix multiplication and transformations depend upon. Develop the habit of writing vectors as columns from the beginning.
v = ( 6 − 2 ) = ( 4 )
( 1 − 5 ) ( −4 )
| Type | Description | Formula |
|---|---|---|
| Position vector | Location of a point P = (a, b) relative to the origin | p = (a, b) |
| Displacement vector | Movement from point A to point B; records change, not location | d = (x₂−x₁, y₂−y₁) |
A position vector describes the location of a point relative to the origin. If P is the point (a, b), its position vector is simply the vector from the origin to P. Position vectors let you use the algebra of vectors to do coordinate geometry.
A displacement vector describes movement from one place to another. It records how much you moved in each direction, not where you started or ended. The same displacement vector could represent many different journeys — any journey that moves the same amount in the same directions.
If a is the position vector of A and b is the position vector of B, then the displacement from A to B is b − a.
When two vectors are combined — two forces acting on the same object, two displacements taken one after another — the result is their sum. The result is called the resultant vector.
Tip-to-Tail Rule: Draw u as an arrow. Place the tail of v at the tip of u. The sum u + v is the arrow from the tail of u to the tip of v.
Parallelogram Rule: Place u and v both starting from the same point. Complete the parallelogram formed by the two arrows. The diagonal of the parallelogram — starting from the shared tail — is u + v.
Vector u is drawn from the origin to point P. Vector v is drawn starting at P and ending at Q. The resultant u + v is the arrow from the origin directly to Q.
Commutativity: u + v = v + u
Associativity: (u + v) + w = u + (v + w)
Identity: v + 0 = v
Inverse: v + (−v) = 0
The geometric construction of vector addition has a direct algebraic equivalent. To add two vectors, add their corresponding components. To subtract, subtract corresponding components.
( u₁ ) + ( v₁ ) = ( u₁ + v₁ )
( u₂ ) ( v₂ ) ( u₂ + v₂ )
( u₁ ) − ( v₁ ) = ( u₁ − v₁ )
( u₂ ) ( v₂ ) ( u₂ − v₂ )
d = ( 5 ) + ( −1 ) = ( 5 + (−1) ) = ( 4 )
( 2 ) ( 4 ) ( 2 + 4 ) ( 6 )
A scalar is a single real number (as distinguished from a vector). Multiplying a vector by a scalar changes its magnitude and possibly its direction, but preserves the linear structure.
If c is a scalar and v = (v₁, v₂), then cv = (cv₁, cv₂)
c > 1: vector is stretched (longer), same direction
0 < c < 1: vector is compressed (shorter), same direction
c = −1: vector is reversed — same magnitude, opposite direction
c < 0: vector is scaled and reversed
c = 0: result is the zero vector 0
4v = ( 4·3 ) = ( 12 )
( 4·(−2) ) ( −8 )
−v = ( −3 )
( 2 )
½v = ( 3/2 )
( −1 )
For scalars c, d and vectors u, v:
c(u + v) = cu + cv
(c + d)v = cv + dv
(cd)v = c(dv)
1·v = v
The properties of vector addition (Section 2.5) and scalar multiplication listed here form the complete list of vector space axioms. Chapter 8 will formalize this structure and show that polynomials, matrices, and functions satisfy the same rules.
The magnitude of a vector is its length — the distance from tail to tip. It is a scalar (a non-negative number). In two dimensions, the magnitude formula comes directly from the Pythagorean theorem applied to the horizontal and vertical components.
For v = (v₁, v₂):
‖v‖ = √(v₁² + v₂²)
The components v₁ and v₂ form the legs of a right triangle. The magnitude is the hypotenuse.
Properties: ‖v‖ ≥ 0 for all v · ‖v‖ = 0 iff v = 0 · ‖cv‖ = |c|·‖v‖
‖d‖ = √(40² + 30²)
= √(1600 + 900)
= √2500
= 50
A unit vector is a vector with magnitude exactly 1. Unit vectors describe direction without any information about magnitude — they are the "pure directions" of the vector world. To find the unit vector in the direction of v, divide v by its magnitude.
The unit vector in the direction of v is:
v̂ = v / ‖v‖
If you scale v by 1/‖v‖, the result has magnitude ‖v‖/‖v‖ = 1. The hat notation v̂ is standard for unit vectors.
‖v‖ = √(3² + 4²) = √(9 + 16) = √25 = 5
v̂ = (1/5)(3) = (3/5)
(4) (4/5)
The two most important unit vectors in two dimensions are:
e₁ = (1, 0) e₂ = (0, 1)
Also called i and j in physics and engineering. Every two-dimensional vector decomposes as:
(v₁, v₂) = v₁ e₁ + v₂ e₂
This decomposition — writing a vector as a combination of basis vectors — previews Chapter 9's central idea of linear combinations and basis.
A vector in the plane has a direction that can be described by an angle. The direction angle of a vector is the angle θ measured counterclockwise from the positive x-axis to the vector. For v = (v₁, v₂) with v₁ ≠ 0, we have tan θ = v₂/v₁. However, you must use the signs of both components to determine the correct quadrant.
| Sign of v₁ | Sign of v₂ | Quadrant | Angle range |
|---|---|---|---|
| Positive | Positive | I | 0° to 90° |
| Negative | Positive | II | 90° to 180° — add 180° to arctan result |
| Negative | Negative | III | 180° to 270° — add 180° to arctan result |
| Positive | Negative | IV | 270° to 360° — angle is negative or add 360° |
If you know a vector's magnitude r = ‖v‖ and direction angle θ, its components are:
v = (r cos θ, r sin θ)
This connects component form to rectangular–polar coordinate conversion from precalculus.
tan θ = 30/40 = 3/4
θ = arctan(3/4) ≈ 36.87°
F = ( 10 cos(120°) ) = ( 10·(−1/2) ) = ( −5 )
( 10 sin(120°) ) ( 10·(√3/2) ) ( 5√3 )
Everything developed so far for two-dimensional vectors extends naturally to three dimensions. Three-dimensional space requires a third coordinate axis — typically called the z-axis — perpendicular to both the x-axis and the y-axis.
A vector in three-dimensional space has three components:
v = ( v₁ )
( v₂ )
( v₃ )
Addition: component by component in all three coordinates.
Scalar multiplication: c·(v₁, v₂, v₃) = (cv₁, cv₂, cv₃).
Magnitude: ‖v‖ = √(v₁² + v₂² + v₃²)
Standard basis: e₁ = (1,0,0), e₂ = (0,1,0), e₃ = (0,0,1) [also written i, j, k]
Apply the Pythagorean theorem twice. In the xy-plane, the horizontal distance is √(v₁² + v₂²). Then use that horizontal distance and v₃ as legs of a second right triangle: √((√(v₁² + v₂²))² + v₃²) = √(v₁² + v₂² + v₃²).
‖v‖ = √(2² + (−3)² + 6²)
= √(4 + 9 + 36)
= √49
= 7
u + v = ( 1+3 ) = ( 4 )
( 0+(−1) ) ( −1 )
( −2+4 ) ( 2 )
2u = ( 2 ) so 2u − v = ( 2−3 ) = ( −1 )
( 0 ) ( 0−(−1) ) ( 1 )
( −4 ) ( −4−4 ) ( −8 )
When a traveler or vehicle moves from one location to another, the displacement is a vector. GPS systems represent position as vectors in three dimensions (latitude, longitude, altitude). Navigation problems involving multiple legs of a journey use vector addition.
Example. An aircraft flies (300, 0) km east, then (0, 400) km north. Its total displacement is (300, 400) km, and its distance from the origin is √(300² + 400²) = 500 km.
In physics, a force is a vector — it has both magnitude (strength, measured in newtons or pounds) and direction. When multiple forces act on an object, the net force is the vector sum of all individual forces. This principle, called the superposition of forces, is one of the most important in mechanics.
Example. Two people push a cart: F₁ = (20, 0) N east; F₂ = (0, 15) N north. Net force: F₁ + F₂ = (20, 15) N, with magnitude √(400 + 225) = 25 N.
A velocity is a vector describing both speed and direction of motion. Distinguishing speed (scalar) from velocity (vector) is an important step in physics readiness.
Example. A boat moves with velocity (6, 0) km/h relative to the water. A river current has velocity (0, 2) km/h. The boat's ground velocity is (6, 2) km/h with speed √(36 + 4) = √40 ≈ 6.32 km/h.
In computer graphics, every object is described by the positions of its vertices — points in two or three dimensions. Transformations (rotation, scaling, translation, reflection) act on these position vectors. The entire pipeline from 3D scene to 2D screen image is built on vector arithmetic.
Example. A triangle has vertices at (0, 0), (4, 0), and (2, 3). Translating by the vector (5, 1) moves every vertex: the new vertices are (5, 1), (9, 1), and (7, 4).
In data science and machine learning, a data record — say, the height, weight, age, and income of a person — is naturally represented as a vector. Operations on these data vectors — distances between records, projections onto directions of variation, combinations — are all linear algebra operations. Chapter 11 (orthogonality and projections) and Chapter 14 (applications) explore these ideas in depth.
Correct: ‖(3, 4)‖ = √(3² + 4²) = √25 = 5. Always square, sum, and take the square root.
Correct: Add the component vectors first, then compute the magnitude of the sum. The triangle inequality only guarantees ‖u + v‖ ≤ ‖u‖ + ‖v‖.
Correct: The direction angle is computed from the components using trigonometry. It is not a component. Components are rectangular coordinates v₁ and v₂.
Correct: (3, 4) as a column vector and [3 4] as a row vector are different objects in linear algebra. Matrix multiplication and transformations depend critically on this distinction.
Correct: v + (−v) = 0 (bold zero), the zero vector (0, 0). It is not the same as the scalar 0.
Correct: Vector A→B = b − a. Vector B→A = a − b = −(b − a). Same magnitude, opposite directions. They are negatives, not equal.
Correct: ‖(2, 6, 3)‖ = √(4 + 36 + 9) = √49 = 7. All three components must be included.
Correct: ‖(6, 8)‖ = 10, so v̂ = (6/10, 8/10) = (3/5, 4/5). Always verify the result has magnitude 1.
Vectors are the starting point of linear algebra, but they quickly lead to larger structures.
From a single vector to a system. A single vector describes one quantity with direction. When multiple vectors must satisfy conditions simultaneously — the forces on a structure must balance, the flows in a network must be consistent, the coordinates of an image must satisfy transformation rules — you need a system of equations. Chapter 3 develops the theory of systems of linear equations and connects them back to the vector language developed here.
Vectors as columns of a matrix. When you collect several vectors side by side, you get a matrix. Chapter 4 develops matrix operations, and later chapters show that the relationship between a matrix and its columns is central to understanding rank, null space, and dimension.
Vectors as inputs and outputs of transformations. Chapter 7 introduces linear transformations — functions that take vectors as inputs and produce vectors as outputs. These generalize the geometric transformations from precalculus to arbitrary dimensions.
Vectors in higher dimensions. The ideas of this chapter extend to ℝⁿ — vectors with any number of components. A data record with 100 features is a vector in ℝ¹⁰⁰. The same formulas for addition, scalar multiplication, magnitude, and unit vectors apply. Chapter 8 formally introduces the concept of a vector space.
Write each vector using column vector notation.
Let u = (2, −5), v = (−3, 4), w = (1, 0). Compute each of the following.
Let a = (1, −2, 3), b = (0, 4, −1). Compute:
Compute the magnitude of each vector, then find the unit vector in the same direction.
Each problem contains a mistake. Identify the error and give the correct result.
What is the correct magnitude? What formula should be used?
Explain the error. What is the correct vector?
What went wrong with the second component? What is the correct answer?
Identify all errors. Compute the correct unit vector and verify its magnitude.
Using u = (3, 0) and v = (−3, 0), show this claim is false by computing both sides.
What component was forgotten? What is the correct magnitude?
Chapter 3 — Systems of Linear Equations. When several linear conditions must hold simultaneously, the unknowns can be organized into a vector x, and the conditions expressed using matrix-vector equations like Ax = b. Solving a system means finding the vector x that satisfies all conditions at once.
Chapter 7 — Linear Transformations. A linear transformation takes a vector as input and produces a vector as output. The geometric transformations of this chapter — stretching, rotating, reflecting — are all linear transformations. Chapter 7 develops the full theory.
Chapter 8 — Vector Spaces. The rules satisfied by vectors in this chapter become the axioms of a vector space. Spaces of polynomials, matrices, and functions satisfy the same rules. Chapter 8 shows how far the vector idea extends.
In Calculus. The gradient of a function is a vector. The derivative of a vector-valued curve is a vector. Tangent planes use normal vectors. The Jacobian matrix collects partial derivatives of vector-valued functions. Every idea in this chapter prepares students for the multivariable world.
In Physics, Engineering, and Data Science. Velocity, acceleration, force, torque, electric field, and magnetic field are all vectors. In machine learning, a dataset is a collection of high-dimensional vectors. Distance, similarity, projection, and optimization algorithms all rely on vector arithmetic and geometry.
Answers may be placed in the answer key.