Geometry Basics: Points, Lines, and Planes
Table of Contents
Introduction to Geometry
Geometry is the branch of mathematics that deals with the properties, measurement, and relationships of points, lines, angles, surfaces, and solids. The word "geometry" comes from the Greek words "geo" (earth) and "metron" (measurement), meaning "earth measurement."
The foundational elements of geometry are points, lines, and planes. These are considered undefined terms in geometry because they form the basis for defining all other geometric concepts.
Points
Points are typically represented by dots and labeled with uppercase letters such as A, B, C, etc.
Properties of Points
- A point has position but no dimension
- Points are named using capital letters
- A point represents an exact location
Coordinates of Points
In the Cartesian coordinate system, a point is represented by an ordered pair (x, y) in two dimensions or by an ordered triplet (x, y, z) in three dimensions.
Distance Between Two Points
The distance between two points A(x₁, y₁) and B(x₂, y₂) can be calculated using the distance formula:
d(A, B) = √[9 + 16]
d(A, B) = √25
d(A, B) = 5 units
Lines
Lines are typically represented by a straight line with arrows at both ends, indicating that it extends infinitely. Lines are often labeled with lowercase letters like l, m, n, or by naming two points on the line.
Types of Lines
- Line segment: A part of a line with two endpoints
- Ray: A part of a line with one endpoint, extending infinitely in the other direction
- Parallel lines: Lines in the same plane that never intersect
- Perpendicular lines: Lines that intersect to form right angles (90°)
- Intersecting lines: Lines that cross at exactly one point
Equations of Lines
In coordinate geometry, lines can be represented by equations. The standard forms include:
Where m is the slope and b is the y-intercept
Where m is the slope and (x₁, y₁) is a point on the line
Where A, B, and C are constants, and A and B are not both zero
Finding the Slope of a Line
The slope of a line passing through points A(x₁, y₁) and B(x₂, y₂) is:
m = 4 / 3
m = 1.33
y - 5 = 2(x - 3)
y - 5 = 2x - 6
y = 2x - 1
Relationships Between Lines
Two distinct lines in a plane can either be parallel or intersecting.
The slope of the second line is also 3.
Since the slopes are equal, the lines are parallel.
The slope of the second line is -1/2.
Product of slopes = 2 × (-1/2) = -1
Since the product of the slopes is -1, the lines are perpendicular.
Planes
Planes are typically represented by shapes that look like parallelograms or rectangles and are labeled with italic uppercase letters like P, Q, R, or by naming three non-collinear points within the plane.
Properties of Planes
- A plane extends infinitely in all directions
- A plane is determined by three non-collinear points
- A plane can also be determined by a line and a point not on the line
- A plane can be determined by two intersecting lines
- A plane can be determined by two parallel lines
Equation of a Plane
In three-dimensional coordinate geometry, a plane can be represented by the equation:
Where A, B, C, and D are constants, and A, B, and C are not all zero.
Where (A, B, C) is the normal vector and (x₀, y₀, z₀) is a point on the plane.
2(x - 1) + (-1)(y - 2) + 4(z - 3) = 0
2x - 2 - y + 2 + 4z - 12 = 0
2x - y + 4z - 12 = 0
Distance From a Point to a Plane
The distance from a point (x₀, y₀, z₀) to a plane Ax + By + Cz + D = 0 is given by:
2x - y + 2z - 5 = 0
Now, use the distance formula:
d = |2(2) - 1(3) + 2(4) - 5| / √(2² + (-1)² + 2²)
d = |4 - 3 + 8 - 5| / √(4 + 1 + 4)
d = |4| / √9
d = 4 / 3
d ≈ 1.33 units
Relationships Between Elements
Relationships Between Points and Lines
- A line contains infinitely many points
- Two distinct points determine exactly one line
- A point is either on a line or not on a line
Relationships Between Lines
- Two distinct lines either intersect at exactly one point or are parallel
- If two lines intersect, they determine a plane
- In three-dimensional space, two lines can also be skew (neither parallel nor intersecting)
Relationships Between Lines and Planes
- A line is either contained in a plane, intersects a plane at exactly one point, or is parallel to a plane
- If a line intersects a plane and is not contained in the plane, the intersection is exactly one point
- Two distinct planes either intersect in a line or are parallel
Relationships Between Planes
- Two distinct planes either intersect in a line or are parallel
- Three distinct planes can intersect in a point, a line, or not at all
- A plane contains infinitely many lines
Practice Problems
d(A, B) = √[(7 - 3)² + (5 - (-2))²]
d(A, B) = √[16 + 49]
d(A, B) = √65
d(A, B) ≈ 8.06 units
m = (5 - (-1)) / (-2 - 4)
m = 6 / (-6)
m = -1
y - (-4) = 3(x - 2)
y + 4 = 3x - 6
y = 3x - 10
The slope of the second line is -1/2.
Product of slopes = 2 × (-1/2) = -1
Since the product of the slopes is -1, the lines are perpendicular.
Vector AB = (0-1, 1-0, 3-2) = (-1, 1, 1)
Vector AC = (2-1, 1-0, 1-2) = (1, 1, -1)
Step 2: Find the normal vector by taking the cross product.
Normal vector = AB × AC
= | i j k |
| -1 1 1 |
| 1 1 -1 |
= i(1×(-1) - 1×1) - j((-1)×(-1) - 1×1) + k((-1)×1 - 1×1)
= i(-1-1) - j(1-1) + k(-1-1)
= -2i + 0j - 2k
= (-2, 0, -2)
Step 3: Use the point-normal form with point A(1, 0, 2).
-2(x-1) + 0(y-0) + (-2)(z-2) = 0
-2x+2 - 2z+4 = 0
-2x - 2z + 6 = 0
x + z = 3
Test Your Knowledge
Question 1: The distance between the points (3, 4) and (6, 8) is:
Question 2: The slope of the line passing through the points (2, 3) and (5, 9) is:
Question 3: Which of the following is the equation of a line with slope 2 passing through the point (3, -1)?
Question 4: Two lines are perpendicular if their slopes are:
Question 5: The general form of the equation of a plane in 3D space is: