The Central Idea
A vector is a directed quantity. It has two essential features:
- a magnitude, which tells how much; and
- a direction, which tells which way.
Geometrically, a vector is often represented by an arrow.
The length of the arrow represents magnitude, while the arrowhead indicates direction.
Why Direction Matters
Suppose two people each walk \(5\) meters.
If one walks east and the other walks north, they have traveled the same distance but have not undergone the same displacement.
The number \(5\) alone does not distinguish the two motions.
A vector can:
\[ \text{5 meters east} \]or
\[ \text{5 meters north}. \]The direction is part of the quantity itself.
Scalars and Vectors
A scalar is a quantity described by magnitude alone.
A vector requires magnitude and direction.
| Quantity | Typical Type | What Matters? |
|---|---|---|
| Temperature | Scalar | Amount |
| Mass | Scalar | Amount |
| Elapsed time | Scalar | Amount |
| Displacement | Vector | Amount and direction |
| Velocity | Vector | Amount and direction |
| Force | Vector | Amount and direction |
Distance and Displacement Are Different
Distance records how much path length has been traveled. It is usually treated as a scalar.
Displacement records the directed change from an initial position to a final position. It is a vector.
If you walk \(3\) meters east and then \(3\) meters west, your total distance is
\[ 3+3=6\text{ meters}. \]But your final position is the same as your initial position, so your net displacement is the zero vector.
A Vector Does Not Record the Entire Path
A displacement vector connects a starting position to an ending position. It does not, by itself, record the route taken between them.
Two very different paths can therefore produce the same displacement.
This distinction is important whenever we separate where something ended up from how it got there.
Directed Line Segments
A geometric vector can be drawn as a directed line segment.
If point \(A\) is the initial point and point \(B\) is the terminal point, the vector from \(A\) to \(B\) may be written
\[ \overrightarrow{AB}. \]The order matters.
\[ \overrightarrow{AB} \]points from \(A\) toward \(B\), while
\[ \overrightarrow{BA} \]points in the opposite direction.
Initial and Terminal Points
For the directed segment
\[ \overrightarrow{AB}, \]- \(A\) is the initial point;
- \(B\) is the terminal point.
The arrow points from the initial point to the terminal point.
This simple order convention becomes important when vectors are converted into coordinate form.
Vector Notation
Vectors are written in several common ways.
A bold letter may be used:
\[ \mathbf{v}. \]An arrow may be placed above a letter:
\[ \vec{v}. \]A vector between named points may be written:
\[ \overrightarrow{AB}. \]These notations emphasize the same basic idea: the object is directed.
Magnitude of a Vector
The magnitude of a vector is its size or length.
Common notation includes
\[ \|\mathbf{v}\| \]or, in some contexts,
\[ |\mathbf{v}|. \]The notation
\[ \|\mathbf{v}\| \]is especially useful because it visually distinguishes the vector \(\mathbf{v}\) from its scalar magnitude.
A Vector and Its Magnitude Are Different Objects
The vector
\[ \mathbf{v} \]contains directional information.
Its magnitude
\[ \|\mathbf{v}\| \]is a nonnegative scalar.
Thus,
\[ \mathbf{v}\neq \|\mathbf{v}\| \]in the ordinary sense: one is a vector and the other is a scalar.
Direction of a Vector
Direction tells which way the vector points.
In the plane, direction might be described verbally:
- east;
- northwest;
- downward;
- toward a particular point.
It may also be described numerically using an angle.
Later lessons will develop the connection between direction, components, angles, and unit vectors.
Magnitude and Direction Work Together
Consider two arrows that point east.
If one has length \(2\) and the other has length \(7\), their directions match but their magnitudes do not.
Now consider two arrows of length \(5\), one pointing east and the other west.
Their magnitudes match but their directions do not.
For two nonzero vectors to be equal, both magnitude and direction must agree.
Equivalent Vector Arrows
A vector is not determined by where its arrow happens to be drawn.
Two arrows represent the same vector when they have the same magnitude and the same direction, even if they begin at different locations.
This allows a vector arrow to be translated parallel to itself without changing the vector it represents.
Free Vectors
In elementary vector geometry, we often work with free vectors.
A free vector is characterized by magnitude and direction rather than by a fixed location.
If an arrow is moved without changing its length or direction, it still represents the same free vector.
This is why vector addition can be carried out by moving arrows into a convenient head-to-tail arrangement.
Bound Vectors and Context
Some applications attach a vector to a particular point.
For example, a force may act at a specific point on a rigid object, or a velocity vector may be attached to a specific point on a moving curve.
Such context can make location significant.
In this introductory lesson, however, our default geometric model will be the free vector unless a problem states otherwise.
Equal Vectors
Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are equal when they represent the same directed quantity.
\[ \boxed{\mathbf{u}=\mathbf{v}} \]means that they have equal magnitude and the same direction.
Their drawings may occupy different places in a diagram.
Opposite Vectors
The vector opposite to \(\mathbf{v}\) is written
\[ -\mathbf{v}. \]It has the same magnitude as \(\mathbf{v}\) but points in the opposite direction.
\[ \|-\mathbf{v}\|=\|\mathbf{v}\|. \]Geometrically, reversing the arrow reverses the vector.
Reversing Endpoints Reverses the Vector
If
\[ \mathbf{v}=\overrightarrow{AB}, \]then
\[ \overrightarrow{BA}=-\mathbf{v}. \]Therefore,
\[ \boxed{\overrightarrow{BA}=-\overrightarrow{AB}}. \]This is one of the simplest ways to see what a negative vector means.
The Zero Vector
The zero vector is written
\[ \mathbf{0}. \]Its magnitude is
\[ \|\mathbf{0}\|=0. \]It represents no net displacement or no directed change.
Does the Zero Vector Have a Direction?
A nonzero vector has a definite direction.
The zero vector is different. Because its magnitude is \(0\), there is no arrow length available to distinguish one direction from another.
For that reason, the zero vector is normally treated as having no defined direction.
Vectors on a Number Line
One-dimensional vectors provide a useful bridge from the number line to higher-dimensional geometry.
On a horizontal number line:
- a positive directed change can be represented by an arrow to the right;
- a negative directed change can be represented by an arrow to the left.
In one dimension, sign can encode the two possible directions.
Example: One-Dimensional Displacement
Suppose a point moves from \(x=2\) to \(x=7\).
The directed change is
\[ 7-2=5. \]The displacement has magnitude \(5\) and points in the positive direction.
If the motion is reversed, from \(7\) to \(2\), the directed change is
\[ 2-7=-5. \]The magnitude is still \(5\), but the direction is reversed.
Vectors in the Coordinate Plane
In two dimensions, a vector can be represented by horizontal and vertical changes.
A common component notation is
\[ \langle a,b\rangle. \]Here:
- \(a\) records horizontal directed change;
- \(b\) records vertical directed change.
The next lesson develops components, magnitude, direction, and unit vectors systematically.
Reading a Simple Component Vector
Consider
\[ \mathbf{v}=\langle 3,2\rangle. \]One geometric interpretation is:
- move \(3\) units to the right;
- move \(2\) units upward.
Together these changes determine a single directed displacement.
Negative Components Encode Direction
Consider
\[ \mathbf{v}=\langle -4,3\rangle. \]The negative first component indicates \(4\) units in the negative horizontal direction, while the positive second component indicates \(3\) units in the positive vertical direction.
Thus, component signs help encode spatial direction relative to the chosen coordinate axes.
From Two Points to a Vector
Suppose
\[ A=(x_1,y_1) \]and
\[ B=(x_2,y_2). \]The vector from \(A\) to \(B\) is obtained by subtracting initial coordinates from terminal coordinates:
\[ \boxed{\overrightarrow{AB} =\langle x_2-x_1,\;y_2-y_1\rangle}. \]The order follows the direction of the arrow: terminal minus initial.
Example: Vector Between Two Points
Let
\[ A=(2,1) \]and
\[ B=(7,4). \]Then
\[ \overrightarrow{AB} =\langle 7-2,\;4-1\rangle =\boxed{\langle 5,3\rangle}. \]The displacement is \(5\) units right and \(3\) units up.
Reversing the Same Example
Using
\[ A=(2,1),\qquad B=(7,4), \]the reverse vector is
\[ \overrightarrow{BA} =\langle 2-7,\;1-4\rangle =\langle -5,-3\rangle. \]Therefore,
\[ \boxed{\overrightarrow{BA} =-\overrightarrow{AB}}. \]Position Vectors
A point can be associated with a vector beginning at the origin.
If
\[ P=(a,b), \]then the vector from the origin \(O=(0,0)\) to \(P\) is
\[ \overrightarrow{OP}=\langle a,b\rangle. \]This is called a position vector of the point \(P\) relative to the chosen origin.
Points and Vectors Are Related but Not Identical
The ordered pair
\[ (3,2) \]can name a point in the coordinate plane.
The notation
\[ \langle 3,2\rangle \]can represent a vector.
The numerical entries may match, but the mathematical roles differ: a point represents a location, while a vector represents a directed displacement or directed quantity.
The Same Vector Can Connect Many Pairs of Points
Consider
\[ \overrightarrow{AB}=\langle 4,1\rangle. \]If another pair of points \(C\) and \(D\) also satisfies
\[ \overrightarrow{CD}=\langle 4,1\rangle, \]then
\[ \overrightarrow{AB}=\overrightarrow{CD}. \]The arrows may be drawn in different places, but they represent the same free vector.
Example: Comparing Two Directed Segments
Let
\[ A=(1,2),\qquad B=(5,5), \]and
\[ C=(-2,4),\qquad D=(2,7). \]Then
\[ \overrightarrow{AB} =\langle 5-1,\;5-2\rangle =\langle 4,3\rangle, \]while
\[ \overrightarrow{CD} =\langle 2-(-2),\;7-4\rangle =\langle 4,3\rangle. \]Therefore,
\[ \boxed{\overrightarrow{AB}=\overrightarrow{CD}}. \]Translation as a Vector Idea
A geometric translation moves every point by the same directed amount.
That directed amount can be represented by a vector.
For example, the translation
\[ \langle 4,-2\rangle \]moves each point \(4\) units right and \(2\) units down.
Vectors therefore provide an algebraic language for translations.
Vectors Connect Geometry and Algebra
A geometric arrow and an algebraic component list can describe the same vector.
This dual viewpoint is powerful:
- geometry makes magnitude and direction visible;
- algebra makes computation systematic.
Much of vector mathematics consists of moving fluently between these two representations.
Vectors in Three Dimensions
In three-dimensional space, a vector may be written
\[ \langle a,b,c\rangle. \]The three components describe directed change along three coordinate axes.
For example,
\[ \langle 2,-1,4\rangle \]describes a directed change involving all three dimensions.
Vectors Are Not Limited to Two or Three Dimensions
The arrow picture is easiest to visualize in ordinary geometric space, but the algebraic idea extends beyond three dimensions.
A vector such as
\[ \langle x_1,x_2,\ldots,x_n\rangle \]can have \(n\) components.
Higher-dimensional vectors are important in linear algebra, data analysis, optimization, physics, computing, and many other fields.
Physical Vectors Often Carry Units
A displacement vector may be measured in meters.
A velocity vector may be measured in meters per second.
A force vector may be measured in newtons.
The vector structure supplies direction, but the underlying quantity may still require physical units.
Example: Velocity as a Directed Quantity
A speed of
\[ 60\text{ km/h} \]gives a magnitude.
A velocity such as
\[ 60\text{ km/h east} \]supplies both magnitude and direction.
Changing only the direction changes the velocity vector even if the speed remains \(60\text{ km/h}\).
Example: Force as a Vector
Suppose two forces each have magnitude \(10\) newtons.
If one acts upward and the other acts downward, the magnitudes are equal but the vectors are opposite.
Symbolically, if the upward force is \(\mathbf{F}\), the downward force can be represented as
\[ -\mathbf{F}. \]Vector Diagrams Depend on Scale
A drawn arrow is a model.
If a diagram uses the scale
\[ 1\text{ cm}=5\text{ units}, \]then an arrow \(4\) cm long represents a vector of magnitude
\[ 4(5)=20\text{ units}. \]The numerical vector is not literally the ink length on the page; the drawing encodes the vector according to a chosen scale.
Direction Depends on the Reference System
Words such as right, left, north, upward, or clockwise assume a frame of reference.
Component notation also depends on chosen coordinate axes.
The vector represents a directed relationship, but its numerical description depends on how the space is coordinated.
Later lessons on coordinate changes and bases will formalize this idea.
Vectors Can Represent Change Between States
The geometric picture suggests a broader interpretation.
If one state is represented by a point and another state by a second point, a vector can describe the directed change between them.
This is one reason vector methods extend naturally from physical space to higher-dimensional mathematical models.
The Arrow Is a Representation, Not the Vector Itself
A vector can be represented by an arrow, components, symbols, coordinates, or equations.
These are representations of the underlying mathematical object.
Just as the same number can be written in several forms, the same vector can be expressed in several equivalent ways.
Vector Equality in Component Form
If
\[ \mathbf{u}=\langle a,b\rangle \]and
\[ \mathbf{v}=\langle c,d\rangle, \]then the vectors are equal exactly when corresponding components are equal:
\[ \boxed{\mathbf{u}=\mathbf{v} \iff a=c\text{ and }b=d}. \]This gives an algebraic test for geometric vector equality.
Example: Testing Equality
Compare
\[ \mathbf{u}=\langle 3,-2\rangle \]and
\[ \mathbf{v}=\langle 3,-2\rangle. \]Since both corresponding components agree,
\[ \boxed{\mathbf{u}=\mathbf{v}}. \]Their arrows may still be drawn from different initial points.
Example: Same Magnitude Does Not Guarantee Equality
Consider
\[ \mathbf{u}=\langle 3,0\rangle \]and
\[ \mathbf{v}=\langle -3,0\rangle. \]Both have magnitude \(3\), but they point in opposite directions.
Therefore,
\[ \boxed{\mathbf{u}\neq\mathbf{v}}. \]Example: Same Direction Does Not Guarantee Equality
Consider two vectors that point east, one with magnitude \(2\) and one with magnitude \(6\).
Their directions agree, but their magnitudes do not.
Therefore, they are not equal vectors.
Equality requires both pieces of information.
Parallel Vectors
Vectors whose directions lie along parallel lines are often described as parallel vectors.
They may point in the same direction or in opposite directions.
Later, scalar multiplication will make this relationship algebraic: one vector is parallel to another when it is a scalar multiple of the other, subject to the usual treatment of the zero vector.
Collinearity and Directed Change
If points \(A\), \(B\), and \(C\) lie on one line, vectors such as
\[ \overrightarrow{AB} \]and
\[ \overrightarrow{AC} \]point along the same line.
Their directions and magnitudes depend on the order and spacing of the points.
Vectors and Rays Are Not the Same
A ray begins at a specific endpoint and extends infinitely in one direction.
A vector has a finite magnitude.
An arrow used to draw a vector resembles a ray visually, but the two objects play different mathematical roles.
Vectors and Line Segments Are Not the Same
An ordinary line segment has endpoints and a length but no preferred direction.
A directed line segment distinguishes an initial point from a terminal point.
That direction is what makes the directed segment suitable for representing a vector.
Vectors and Coordinates Are Not the Same
Coordinates describe locations relative to a coordinate system.
Vector components describe directed changes relative to coordinate axes.
Their numerical formats may look similar, but the distinction between location and directed change should remain clear.
Common Error: Treating a Vector as Only a Length
If a problem says a vector has magnitude \(8\), that information does not determine the vector by itself.
Many vectors can have magnitude \(8\) while pointing in different directions.
Magnitude is only one part of a nonzero vector.
Common Error: Ignoring Endpoint Order
For two points \(A\) and \(B\),
\[ \overrightarrow{AB} \]is not generally the same as
\[ \overrightarrow{BA}. \]Instead,
\[ \boxed{\overrightarrow{BA} =-\overrightarrow{AB}}. \]Reversing the endpoints reverses the direction.
Common Error: Terminal Minus Initial in the Wrong Order
To find
\[ \overrightarrow{AB}, \]subtract the coordinates of \(A\) from the coordinates of \(B\):
\[ \boxed{B-A}. \]Using \(A-B\) produces the opposite vector.
Common Error: Confusing Position with Displacement
The point
\[ P=(4,1) \]describes a location.
The vector
\[ \langle 4,1\rangle \]describes a directed quantity.
They are closely related through the position vector from the origin, but they are not conceptually identical.
Common Error: Thinking a Vector Has a Fixed Drawing Location
For free vectors, moving an arrow without rotating or resizing it does not create a new vector.
The position of the drawing can change while the represented magnitude and direction remain unchanged.
Common Error: Assigning a Direction to the Zero Vector
The zero vector has magnitude \(0\).
Because it has no nonzero arrow length, its direction is not defined in the ordinary sense.
Avoid describing \(\mathbf{0}\) as pointing north, east, or along a particular angle.
A Reliable Procedure for Reading a Vector Diagram
- Identify the initial point.
- Identify the terminal point.
- Observe the direction of the arrow.
- Determine or compare its magnitude.
- Use the stated scale if the diagram has one.
- If coordinates are given, compute terminal minus initial.
- Distinguish the vector from the points used to draw it.
A Reliable Procedure for Finding a Vector from Two Points
If
\[ A=(x_1,y_1),\qquad B=(x_2,y_2), \]and you need \(\overrightarrow{AB}\):
- Start with the terminal point \(B\).
- Subtract the corresponding coordinates of the initial point \(A\).
- Keep the component order consistent.
- Write the result as a vector.
- Check that its signs agree with the geometric direction.
Example: Full Two-Point Workflow
Let
\[ A=(-3,2),\qquad B=(4,-1). \]Compute terminal minus initial:
\[ \overrightarrow{AB} =\langle 4-(-3),\;-1-2\rangle. \]Simplify:
\[ \boxed{\overrightarrow{AB}=\langle 7,-3\rangle}. \]The signs agree with the picture we would expect: \(7\) units right and \(3\) units down.
Checking a Coordinate Vector Geometrically
Suppose
\[ \overrightarrow{AB}=\langle 7,-3\rangle. \]If \(A=(-3,2)\), move \(7\) units right:
\[ -3+7=4. \]Then move \(3\) units down:
\[ 2-3=-1. \]This reaches
\[ B=(4,-1), \]confirming the vector.
Checking by Reversing the Vector
If
\[ \overrightarrow{AB}=\langle 7,-3\rangle, \]then
\[ \overrightarrow{BA}=\langle -7,3\rangle. \]The components change sign because the direction is reversed.
This provides another quick consistency check.
Why Vectors Matter for Spatial Algebra
Vectors turn spatial relationships into objects that can be manipulated algebraically.
Once vectors are represented by components, we can:
- add directed changes;
- subtract directed changes;
- scale vectors;
- measure angles between vectors;
- describe lines and planes;
- apply transformations using matrices.
These operations form the backbone of the rest of this section.
Vectors as Inputs to Transformations
Earlier transformation geometry described motions such as translations, reflections, rotations, and scalings.
Vector language gives those transformations an algebraic input-output structure.
Later in this section, matrices will act on vectors to perform many of these transformations systematically.
Practice 1: Scalar or Vector?
Classify each quantity as scalar or vector.
- \(12\) kilograms of mass
- \(20\) meters east of a starting point
- \(15\) seconds of elapsed time
- a \(40\)-newton force directed upward
Practice 2: Initial and Terminal Points
In
\[ \overrightarrow{PQ}, \]identify the initial point and the terminal point.
Practice 3: Reverse a Vector
If
\[ \mathbf{v}=\overrightarrow{AB}, \]express \(\overrightarrow{BA}\) in terms of \(\mathbf{v}\).
Practice 4: Same Vector or Not?
Two arrows have the same length and point in the same direction, but one is drawn in the upper-left part of a diagram and the other in the lower-right.
Do they represent the same free vector?
Practice 5: Find a Vector from Two Points
Let
\[ A=(1,-2),\qquad B=(6,4). \]Find
\[ \overrightarrow{AB}. \]Practice 6: Reverse the Previous Vector
Using the same points
\[ A=(1,-2),\qquad B=(6,4), \]find
\[ \overrightarrow{BA}. \]Practice 7: Compare Directed Segments
Let
\[ A=(0,1),\qquad B=(3,5), \]and
\[ C=(-4,2),\qquad D=(-1,6). \]Determine whether
\[ \overrightarrow{AB}=\overrightarrow{CD}. \]Practice 8: Interpret Components
Describe the directed movement represented by
\[ \langle -5,2\rangle. \]Practice 9: Position Vector
Find the position vector from the origin to
\[ P=(-2,7). \]Practice 10: Identify the Error
A student wants to find the vector from
\[ A=(2,3) \]to
\[ B=(8,5) \]and writes
\[ \overrightarrow{AB} =\langle 2-8,\;3-5\rangle =\langle -6,-2\rangle. \]Identify and correct the error.
Practice Answers
Practice 1
- Scalar
- Vector
- Scalar
- Vector
Practice 2
In \(\overrightarrow{PQ}\), \(P\) is the initial point and \(Q\) is the terminal point.
Practice 3
\[ \boxed{\overrightarrow{BA}=-\mathbf{v}}. \]Practice 4
Yes. Equal free vectors may be drawn in different locations. Equal magnitude and equal direction are what matter.
Practice 5
\[ \overrightarrow{AB} =\langle 6-1,\;4-(-2)\rangle =\boxed{\langle 5,6\rangle}. \]Practice 6
\[ \overrightarrow{BA} =\langle 1-6,\;-2-4\rangle =\boxed{\langle -5,-6\rangle}. \]As expected,
\[ \overrightarrow{BA}=-\overrightarrow{AB}. \]Practice 7
\[ \overrightarrow{AB} =\langle 3-0,\;5-1\rangle =\langle 3,4\rangle, \]and
\[ \overrightarrow{CD} =\langle -1-(-4),\;6-2\rangle =\langle 3,4\rangle. \]Therefore,
\[ \boxed{\overrightarrow{AB}=\overrightarrow{CD}}. \]Practice 8
The vector represents \(5\) units in the negative horizontal direction and \(2\) units in the positive vertical direction: \(5\) units left and \(2\) units up in the standard coordinate plane.
Practice 9
With \(O=(0,0)\),
\[ \overrightarrow{OP} =\langle -2-0,\;7-0\rangle =\boxed{\langle -2,7\rangle}. \]Practice 10
The subtraction order was reversed. For \(\overrightarrow{AB}\), subtract the initial coordinates from the terminal coordinates:
\[ \overrightarrow{AB} =\langle 8-2,\;5-3\rangle =\boxed{\langle 6,2\rangle}. \]What to Remember
- A vector is a directed quantity with magnitude and direction.
- A scalar has magnitude without directional information.
- A vector can be represented geometrically by a directed line segment.
- For \(\overrightarrow{AB}\), \(A\) is initial and \(B\) is terminal.
- Equal free vectors have the same magnitude and direction.
- Opposite vectors have equal magnitude and opposite direction.
- The zero vector has magnitude \(0\) and no defined direction.
- Coordinate vectors encode directed change through components.
- To find \(\overrightarrow{AB}\), compute terminal minus initial.
- Points describe locations; vectors describe directed quantities.
Where We Go Next
We now have the geometric idea of a vector: magnitude together with direction.
The next lesson, Components, Magnitude, Direction, and Unit Vectors, makes this description computational. We will learn how components encode a vector, how to calculate its magnitude, how angles describe its direction, and how unit vectors isolate direction from size.