The Central Idea
These four relationships answer different spatial questions.
- Parallel: Do two lines or directions run in the same direction without meeting?
- Perpendicular: Do two lines, directions, or surfaces meet at a right angle?
- Collinear: Do several points lie on the same line?
- Coplanar: Do several points or other objects lie in the same plane?
Parallelism and perpendicularity primarily compare directions. Collinearity and coplanarity primarily compare containment in a common geometric object.
Why These Relationships Matter
These relationships appear throughout geometry and beyond.
They are used to:
- classify figures;
- construct coordinate systems;
- analyze slopes;
- describe three-dimensional structures;
- prove geometric properties;
- build perpendicular distances and projections;
- analyze vectors and transformations;
- model engineering and physical systems.
Much of geometry depends on recognizing these relationships correctly.
Parallel Lines
In Euclidean geometry, two distinct coplanar lines are parallel when they do not intersect.
If lines \(\ell\) and \(m\) are parallel, we write
\[ \boxed{\ell\parallel m}. \]The requirement that the lines be coplanar is essential.
Parallel Lines Share a Direction
Parallel lines have the same direction, even though they occupy different positions.
Translating one parallel line perpendicular to itself can move it onto the other.
This connects parallelism to the distinction between position and direction: the positions differ, but the directional structure agrees.
Distinct Parallel Lines Do Not Intersect
If distinct lines \(\ell\) and \(m\) are parallel in a Euclidean plane, then
\[ \boxed{\ell\cap m=\varnothing}. \]Parallelism is therefore a special kind of nonintersection together with coplanarity and matching direction.
Merely failing to intersect is not enough in three-dimensional space.
Parallel Lines and Slope
In the Cartesian plane, nonvertical lines are parallel when they have the same slope and are distinct.
If
\[ \ell_1:y=m_1x+b_1 \]and
\[ \ell_2:y=m_2x+b_2, \]then distinct lines are parallel when
\[ \boxed{m_1=m_2} \]and their intercepts differ.
Example: Parallel Lines from Equations
Consider
\[ y=3x+2 \]and
\[ y=3x-5. \]Both lines have slope \(3\), but their \(y\)-intercepts differ.
Therefore,
\[ \boxed{\ell_1\parallel\ell_2}. \]Vertical Lines Can Be Parallel
Vertical lines do not have finite numerical slopes, but distinct vertical lines are parallel.
For example,
\[ x=2 \]and
\[ x=-4 \]never meet and lie in the same coordinate plane.
Therefore, they are parallel.
Parallel Direction Vectors
Vectors are parallel when one is a scalar multiple of the other.
For vectors
\[ \mathbf{u} \]and
\[ \mathbf{v}, \]parallel direction occurs when
\[ \boxed{\mathbf{v}=k\mathbf{u}} \]for some nonzero scalar \(k\).
Same and Opposite Vector Directions
If
\[ \mathbf{v}=k\mathbf{u} \]with \(k>0\), the vectors point in the same direction.
If \(k<0\), they point in opposite directions.
Both cases represent parallel directional structure.
Example: Parallel Vectors
Let
\[ \mathbf{u}=\langle 2,3\rangle \]and
\[ \mathbf{v}=\langle 6,9\rangle. \]Since
\[ \mathbf{v}=3\mathbf{u}, \]the vectors are parallel.
Parallel Planes
In three-dimensional Euclidean space, distinct planes are parallel when they do not intersect.
Parallel planes share the same orientation while occupying different positions.
For example,
\[ z=2 \]and
\[ z=7 \]are parallel horizontal planes.
Parallel Planes and Normal Vectors
A plane
\[ ax+by+cz=d \]has a normal vector
\[ \mathbf{n}=\langle a,b,c\rangle. \]Two planes are parallel when their normal vectors are parallel.
Thus, if
\[ \mathbf{n}_2=k\mathbf{n}_1, \]the planes have matching orientation. They are either parallel distinct planes or the same plane.
Perpendicular Lines
Two lines are perpendicular when they intersect to form right angles.
If lines \(\ell\) and \(m\) are perpendicular, we write
\[ \boxed{\ell\perp m}. \]Their angle of intersection is
\[ 90^\circ. \]One Right Angle Creates Four Right Angles
When two lines intersect, they form four angles.
If one of those angles is a right angle, then all four are right angles.
Therefore, identifying one \(90^\circ\) angle is enough to establish that the two lines are perpendicular.
Perpendicular Lines and Slopes
For nonvertical and nonhorizontal lines in the coordinate plane, perpendicular slopes are negative reciprocals.
If the slopes are \(m_1\) and \(m_2\), then
\[ \boxed{m_1m_2=-1}. \]Equivalently,
\[ m_2=-\frac{1}{m_1}. \]Example: Perpendicular Slopes
Consider
\[ y=2x+1 \]and
\[ y=-\frac{1}{2}x+4. \]Their slopes satisfy
\[ 2\left(-\frac12\right)=-1. \]Therefore, the lines are perpendicular.
Horizontal and Vertical Lines Are Perpendicular
The negative-reciprocal slope test does not directly apply when one line is vertical because its slope is undefined.
Nevertheless, every horizontal line is perpendicular to every vertical line.
For example,
\[ y=3 \]is perpendicular to
\[ x=5. \]Perpendicular Vectors
Two nonzero vectors are perpendicular when their dot product is zero.
If
\[ \mathbf{u}=\langle u_1,u_2\rangle \]and
\[ \mathbf{v}=\langle v_1,v_2\rangle, \]then
\[ \mathbf{u}\cdot\mathbf{v} = u_1v_1+u_2v_2. \]Perpendicularity occurs when
\[ \boxed{\mathbf{u}\cdot\mathbf{v}=0}. \]Example: Perpendicular Vectors
Let
\[ \mathbf{u}=\langle 3,4\rangle \]and
\[ \mathbf{v}=\langle -4,3\rangle. \]Their dot product is
\[ 3(-4)+4(3)=-12+12=0. \]Therefore,
\[ \boxed{\mathbf{u}\perp\mathbf{v}}. \]Why the Dot Product Detects Perpendicularity
The dot product also satisfies
\[ \mathbf{u}\cdot\mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta, \]where \(\theta\) is the angle between the vectors.
If
\[ \theta=90^\circ, \]then
\[ \cos90^\circ=0, \]so the dot product is zero.
Perpendicular Lines in Three Dimensions
In three-dimensional geometry, two lines are perpendicular when they intersect and their direction vectors are perpendicular.
The intersection requirement matters.
Two lines can have perpendicular direction vectors yet lie in different parts of space and never meet.
In that situation, it is safer to say their directions are orthogonal rather than calling the lines themselves perpendicular.
A Line Perpendicular to a Plane
A line is perpendicular to a plane when it meets the plane and is perpendicular to every line in the plane through the intersection point.
In vector language, a line perpendicular to a plane has a direction vector parallel to a normal vector of the plane.
This gives an efficient coordinate test in three dimensions.
Perpendicular Planes
Two planes are perpendicular when the angle between them is a right angle.
Their normal vectors are then perpendicular.
If
\[ \mathbf{n}_1\cdot\mathbf{n}_2=0, \]the planes are perpendicular.
Collinear Points
Points are collinear when they lie on the same line.
If points \(A\), \(B\), and \(C\) all belong to line \(\ell\), then
\[ A,B,C\in\ell. \]We then say that \(A\), \(B\), and \(C\) are collinear.
Two Points Are Always Collinear
Any two distinct points determine a unique line in Euclidean geometry.
Therefore, any pair of distinct points is automatically collinear.
Collinearity becomes informative when three or more points are being compared.
Testing Collinearity with Slope
In the coordinate plane, three points are collinear when the slopes between appropriate pairs agree, provided the slope comparisons are valid.
For
\[ A=(x_1,y_1), \qquad B=(x_2,y_2), \qquad C=(x_3,y_3), \]one may compare
\[ m_{AB} = \frac{y_2-y_1}{x_2-x_1} \]and
\[ m_{AC} = \frac{y_3-y_1}{x_3-x_1}. \]Matching slopes indicate that the points lie on the same line.
Example: Collinearity by Slope
Consider
\[ A=(1,2), \qquad B=(3,6), \qquad C=(5,10). \]Then
\[ m_{AB} = \frac{6-2}{3-1} = 2 \]and
\[ m_{AC} = \frac{10-2}{5-1} = 2. \]Therefore, the three points are collinear.
Vertical Collinearity
The slope test must be handled carefully for vertical lines.
Points
\[ (4,1),\qquad(4,5),\qquad(4,-3) \]are collinear because they all satisfy
\[ x=4. \]Their pairwise slopes are undefined, but their common \(x\)-coordinate identifies the same vertical line.
Testing Collinearity with Vectors
Three points \(A\), \(B\), and \(C\) are collinear when the displacement vectors
\[ \overrightarrow{AB} \]and
\[ \overrightarrow{AC} \]are parallel.
That means
\[ \boxed{ \overrightarrow{AC} = k\overrightarrow{AB} } \]for some scalar \(k\).
A Determinant Test for Collinearity
For three points in the plane,
\[ A=(x_1,y_1), \quad B=(x_2,y_2), \quad C=(x_3,y_3), \]collinearity can be tested by
\[ \boxed{ \begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1 \end{vmatrix} =0 }. \]This test is closely related to the signed area of the triangle formed by the three points.
Zero Triangle Area Means Collinear
Three noncollinear points form a triangle with positive area.
If the three points lie on one line, the triangle collapses and has area \(0\).
Thus, the determinant test for collinearity is also an area test.
Coplanar Objects
Points, lines, segments, rays, curves, or other objects are coplanar when they lie in the same plane.
If objects all belong to plane \(\Pi\), then they are coplanar.
Coplanarity is therefore a containment relationship involving a common two-dimensional plane.
Three Points Are Always Coplanar
Any three points in Euclidean space are coplanar.
If the points are noncollinear, they determine a unique plane.
If the points are collinear, infinitely many planes contain the line through them.
Coplanarity becomes genuinely restrictive when four or more points are considered.
Four Points Need Not Be Coplanar
Four points in three-dimensional space may or may not lie in one plane.
For example,
\[ A=(0,0,0), \quad B=(1,0,0), \quad C=(0,1,0), \quad D=(0,0,1) \]are not coplanar.
The first three lie in the plane \(z=0\), while \(D\) does not.
Testing Coplanarity with a Plane Equation
If a plane equation is known, coplanarity can be checked by substitution.
Suppose
\[ \Pi:x+2y-z=4. \]A point \(P=(a,b,c)\) lies in the plane exactly when
\[ a+2b-c=4. \]Several points are coplanar in \(\Pi\) if they all satisfy the same plane equation.
Testing Four Points with Vectors
Four points \(A\), \(B\), \(C\), and \(D\) are coplanar when the three displacement vectors
\[ \overrightarrow{AB}, \qquad \overrightarrow{AC}, \qquad \overrightarrow{AD} \]do not span three independent spatial directions.
One standard test uses the scalar triple product:
\[ \boxed{ \overrightarrow{AB} \cdot \left( \overrightarrow{AC} \times \overrightarrow{AD} \right) =0 }. \]A zero result means the four points are coplanar.
Why the Scalar Triple Product Works
The absolute value of
\[ \overrightarrow{AB} \cdot \left( \overrightarrow{AC} \times \overrightarrow{AD} \right) \]measures the volume of the parallelepiped determined by the three vectors.
If the points are coplanar, that three-dimensional volume collapses to \(0\).
Thus, coplanarity can be understood as zero three-dimensional volume.
Parallel Lines Must Be Coplanar
In Euclidean geometry, distinct parallel lines are required to lie in the same plane.
This distinguishes parallel lines from a different three-dimensional relationship: skew lines.
Skew Lines
Skew lines are lines in three-dimensional space that do not intersect and are not parallel.
They are noncoplanar.
Thus:
- parallel lines are coplanar and nonintersecting;
- skew lines are noncoplanar and nonintersecting.
This is why “never meet” is not a complete definition of parallel lines in three dimensions.
Example of Skew Lines
Imagine one line running along the front bottom edge of a cube and another line running along a rear vertical edge.
The lines do not meet.
Their directions are also different, so they are not parallel.
No single plane contains both lines.
Therefore, they are skew.
Intersecting Lines Are Coplanar
Any two intersecting lines determine a unique plane.
Therefore, intersecting lines are automatically coplanar.
Perpendicular lines are a special case of intersecting lines, so perpendicular lines are also coplanar.
Collinear Points Are Coplanar
If several points lie on one line, then they also lie in every plane that contains that line.
Therefore, collinear points are always coplanar.
The converse is not true: coplanar points need not all lie on one line.
Coplanar Does Not Mean Collinear
Consider the points
\[ A=(0,0), \qquad B=(1,0), \qquad C=(0,1). \]They all lie in the same coordinate plane, so they are coplanar.
But they do not lie on one line, so they are not collinear.
Parallel Does Not Mean Collinear
Distinct parallel lines occupy different positions.
A point on one parallel line and a point on another do not automatically lie on a common line with additional points from both lines.
Parallelism compares lines or directions. Collinearity compares points.
Perpendicularity Does Not Mean Vertical and Horizontal
A familiar perpendicular pair is one horizontal line and one vertical line, but perpendicular lines can have any orientation.
For example, lines with slopes
\[ 2 \]and
\[ -\frac12 \]are perpendicular even though neither is horizontal or vertical.
Parallel and Perpendicular Relationships Survive Translation
Translating a geometric figure preserves directions and angles.
Therefore:
- parallel lines remain parallel after a common translation;
- perpendicular lines remain perpendicular after a common translation.
This is one example of a geometric relationship preserved by rigid motion.
Rotation Preserves Parallelism and Perpendicularity
Rotating an entire configuration turns every direction by the same amount.
Differences between directions remain unchanged.
Therefore, parallel lines remain parallel, and perpendicular lines remain perpendicular, after a common rotation.
Reflection Preserves Parallelism and Perpendicularity
Reflection reverses orientation, but it preserves distances and angle magnitudes.
As a result, it also preserves parallel and perpendicular relationships.
These observations foreshadow the later study of geometric transformations and invariants.
Coordinate Axes Provide a Standard Example
In the Cartesian plane, the coordinate axes satisfy
\[ x=0 \]and
\[ y=0. \]They intersect at the origin and are perpendicular:
\[ \boxed{x\text{-axis}\perp y\text{-axis}}. \]The coordinate system itself is built from a perpendicular relationship.
Rectangles Depend on Parallelism and Perpendicularity
A rectangle has opposite sides parallel and adjacent sides perpendicular.
Thus, its structure combines both relationships.
If its sides are aligned with the coordinate axes, horizontal sides are parallel to one another, vertical sides are parallel to one another, and every horizontal side is perpendicular to every vertical side.
Parallelograms Depend on Parallelism
A parallelogram is characterized by two pairs of opposite parallel sides.
Rectangles, rhombi, and squares are special parallelograms, so parallelism remains part of their structure.
More detailed classification of figures will come later in Spatial Relationships.
Perpendicularity Creates Shortest Distances
The shortest distance from a point to a line is measured along a perpendicular segment.
Likewise, the shortest distance from a point to a plane in Euclidean three-dimensional space follows the plane's normal direction.
This makes perpendicularity central to distance, projection, optimization, and measurement.
Parallelism and Constant Distance
Distinct parallel lines in a Euclidean plane remain a constant perpendicular distance apart.
Similarly, distinct parallel planes in Euclidean three-dimensional space remain a constant perpendicular distance apart.
Constant separation is therefore another useful way to understand parallelism.
Common Error: Defining Parallel Lines Only as Lines That Never Meet
In a plane, this description works for distinct Euclidean lines.
In three dimensions, however, skew lines also never meet.
A complete Euclidean definition requires distinct parallel lines to be coplanar.
Common Error: Forgetting the Intercept Condition
Two equations with the same slope do not necessarily describe two distinct parallel lines.
For example,
\[ y=2x+3 \]and
\[ 2y=4x+6 \]describe the same line.
They are coincident, not distinct parallel lines.
Common Error: Applying the Negative-Reciprocal Rule to Vertical Lines
A vertical line has undefined slope.
Therefore, multiplying slopes is not the correct test for a vertical and horizontal pair.
Use geometry directly: horizontal and vertical lines are perpendicular.
Common Error: Assuming Matching Slopes Always Prove Three Points Are Collinear
Slope comparisons must be made carefully when vertical lines are possible.
If denominators become zero, use common \(x\)-coordinates, a vector test, or a determinant test instead.
Common Error: Assuming Any Four Points Are Coplanar
Any three points are coplanar, but four points need not be.
In three-dimensional space, the fourth point may lie outside the plane determined by the first three noncollinear points.
Common Error: Confusing Orthogonal Directions with Perpendicular Lines
Two direction vectors can have dot product \(0\) even if the corresponding lines do not intersect.
In three dimensions, perpendicular lines must both intersect and meet at a right angle.
Common Error: Trusting Appearance
Lines that look parallel may converge slightly.
Lines that look perpendicular may meet at an angle other than \(90^\circ\).
Points that look collinear may not be exactly aligned.
Use markings, coordinates, equations, vectors, or proof rather than visual appearance alone.
A Reliable Procedure for Parallel and Perpendicular Lines
- Determine whether the objects are lines, vectors, or planes.
- Identify a usable representation: diagram, slope, direction vector, or normal vector.
- For nonvertical planar lines, compare slopes.
- Equal slopes suggest parallel direction.
- Negative reciprocal slopes suggest perpendicular direction.
- For vectors, test scalar multiples for parallelism.
- For vectors, test a zero dot product for perpendicularity.
- Check incidence or coplanarity when the geometric definition requires it.
A Reliable Procedure for Collinearity
- Choose one point as a reference.
- Form displacement vectors to the other points.
- Check whether those vectors are scalar multiples.
- In two dimensions, slope can be used when it is defined.
- For a coordinate test that also handles vertical cases, use a determinant.
- Verify that all points satisfy one common line equation when convenient.
A Reliable Procedure for Coplanarity
- If a plane equation is known, substitute each point.
- If three noncollinear points are given, use them to determine a plane.
- Check whether each additional point satisfies that plane.
- Alternatively, form three displacement vectors from one reference point.
- Use a zero scalar triple product to test four-point coplanarity.
A Compact Comparison
| Relationship | Main idea | Typical test | Notation |
|---|---|---|---|
| Parallel | Same directional structure | Equal slopes or scalar-multiple directions | \(\ell\parallel m\) |
| Perpendicular | Right-angle relationship | \(m_1m_2=-1\) or zero dot product | \(\ell\perp m\) |
| Collinear | Points lie on one line | Equal slopes, parallel displacements, or zero determinant | Usually stated in words |
| Coplanar | Objects lie in one plane | Common plane equation or zero scalar triple product | Usually stated in words |
Practice 1: Parallel Slopes
Determine whether the lines
\[ y=4x+1 \]and
\[ y=4x-7 \]are parallel.
Practice 2: Perpendicular Slopes
Determine whether the lines
\[ y=3x+2 \]and
\[ y=-\frac13x+5 \]are perpendicular.
Practice 3: Parallel Vectors
Determine whether
\[ \mathbf{u}=\langle 2,-5\rangle \]and
\[ \mathbf{v}=\langle -6,15\rangle \]are parallel.
Practice 4: Perpendicular Vectors
Determine whether
\[ \mathbf{u}=\langle 5,2\rangle \]and
\[ \mathbf{v}=\langle -2,5\rangle \]are perpendicular.
Practice 5: Collinearity by Slope
Determine whether
\[ A=(0,1), \qquad B=(2,5), \qquad C=(4,9) \]are collinear.
Practice 6: Vertical Collinearity
Determine whether
\[ A=(-3,1), \qquad B=(-3,7), \qquad C=(-3,-4) \]are collinear.
Practice 7: Coplanarity from a Plane Equation
Determine whether
\[ P=(1,1,2) \]lies in the plane
\[ x+y+z=4. \]Practice 8: Four Points in a Plane
Determine whether the points
\[ A=(0,0,0), \quad B=(1,0,0), \quad C=(0,1,0), \quad D=(2,3,0) \]are coplanar.
Practice 9: Parallel or Skew?
Two lines in three-dimensional space do not intersect. Is that information alone enough to conclude that they are parallel?
Practice 10: Identify the Error
A student says, “Any two lines that never intersect are parallel.”
Explain why this statement fails in three-dimensional space.
Practice Answers
Practice 1
Both lines have slope \(4\) and different intercepts.
Therefore,
\[ \boxed{\ell_1\parallel\ell_2}. \]Practice 2
The product of the slopes is
\[ 3\left(-\frac13\right)=-1. \]Therefore,
\[ \boxed{\ell_1\perp\ell_2}. \]Practice 3
Since
\[ \mathbf{v}=-3\mathbf{u}, \]the vectors are parallel and point in opposite directions.
Practice 4
Compute the dot product:
\[ \mathbf{u}\cdot\mathbf{v} = 5(-2)+2(5) = -10+10 = 0. \]Therefore, the vectors are perpendicular.
Practice 5
The slopes are
\[ m_{AB} = \frac{5-1}{2-0} = 2 \]and
\[ m_{AC} = \frac{9-1}{4-0} = 2. \]Therefore, the points are collinear.
Practice 6
All three points have
\[ x=-3. \]Therefore, they lie on the same vertical line and are collinear.
Practice 7
Substitute the coordinates:
\[ 1+1+2=4. \]The equation is satisfied, so
\[ \boxed{P\in\Pi}. \]Practice 8
Every point has
\[ z=0. \]Therefore, all four points lie in the plane
\[ z=0 \]and are coplanar.
Practice 9
No. The lines may be parallel, or they may be skew.
To conclude that they are parallel, they must also be coplanar and have matching direction.
Practice 10
In three-dimensional space, skew lines do not intersect but are not parallel because they are not coplanar. Nonintersection alone is therefore insufficient.
What to Remember
- Distinct parallel lines are coplanar, share a direction, and do not intersect.
- Nonvertical parallel lines have equal slopes.
- Perpendicular lines intersect at \(90^\circ\).
- For suitable nonvertical lines, perpendicular slopes multiply to \(-1\).
- Parallel vectors are scalar multiples.
- Perpendicular vectors have dot product \(0\).
- Collinear points lie on the same line.
- Coplanar objects lie in the same plane.
- Any two points are collinear, and any three points are coplanar.
- Four or more points need not be coplanar.
- Skew lines are noncoplanar, nonparallel, and nonintersecting.
- Collinear points are automatically coplanar, but coplanar points need not be collinear.
Where We Go Next
Parallelism, perpendicularity, collinearity, and coplanarity describe several of the most important structural relationships among spatial objects. The next step is to describe how objects divide surrounding space into inside, outside, edge, and nearby portions.
The next lesson, Interior, Exterior, Boundary, and Neighborhood, develops the language used to distinguish points inside an object, outside it, on its edge, or locally near a chosen point.