In geometry, points, lines, and planes are the fundamental elements used to describe shapes and positions in space. They are considered undefined terms, meaning they are described using basic ideas rather than precise definitions.
Points
A point in geometry is defined as a location in the space that is uniquely determined by an ordered triplet (x, y, z), where x, y, & z are the distances of the point from the X-axis, Y-axis, and Z-axis, respectively, in the 3-Dimensions and is defined by an ordered pair (x, y) in the 2-Dimensions, where x and y are the distances of the point from the X-axis and Y-axis, respectively.
- It is represented using the dot and is named using the capital English alphabet.
- The figure added below shows a point P in 3-D at a distance of x, y, and z from the X-axis, Y-axis, and Z-axis, respectively.
Collinear and Non-Collinear Points
When three or more points lie on the same straight line, they are called collinear points. If the points do not lie on the same straight line, they are called non-collinear points.
Coplanar and Non-Coplanar Points
When the group of points is present on the same plane, then such types of points are known as coplanar points, and if these points do not lie on the same plane, then such types of points are known as non-coplanar points.
Lines
A line is a set of points extending infinitely in both directions in two-dimensional or three-dimensional geometry and is defined as a set of points in 3D that extends infinitely in both directions. It is the smallest distance between any two points either in 2-D or 3-D space. We represent a line with L and in 3-D space, a line is given using the equation,
L: (x - x1) / l = (y - y1) / m = (z - z1) / n
where:
- (x, y, z) Are the position coordinates of any variable point lying on the line
- (x1, y1, z1) are the position coordinates of a point P lying on the line
- l, m, & n are the direction ratios of the line.
In 3D we can also form a line by the intersection of two non-parallel planes.
Line Segment
A line segment is defined as the finite length of the line that is used to join two points in 2-D and 3-D. It is the shortest distance between two points. A line segment between two points A and B is denoted as AB.
A line has infinite length, whereas a line segment is a part of a line and has finite length.
Midpoint
Midpoint is defined as the point on the line segment that divides the line segment into two equal parts. Suppose we have two points A and B, and the line segment joining these two points is AB, and the point P on the line is called the midpoint if it breaks the line into two equal parts such that,
AP = PB
Thus, P is called the midpoint of line segment AB. The image added below shows the line segment AB with P as the midpoint.
Rays
A ray is defined as a line that has a fixed endpoint in one direction but can be extended to infinity in the other direction. It is of infinite length. We define the ray joining points O and A and extending to infinity towards A as
Intersecting and Parallel Lines
In 2-D any two lines can either meet at some point, or they never meet at any point. The lines that meet at some point are called intersecting lines. The distance between the intersecting lines keeps decreasing as we move toward the point of intersection, and at the point of intersection of these lines, the distance between them becomes zero. When two lines intersect, an angle is formed between them.
Two lines that never meet each other in 2-D planes are called parallel lines. For parallel lines, the distance between them is always constant.
Perpendicular Lines
Intersecting lines that intersect at right angles are called perpendicular lines. The angle between these perpendicular lines is always a right angle, or 90 degrees.
Planes
A plane in three-dimensional (3D) geometry is a surface such that the line segment joining any two points lies completely on it. It is the collection of all the points and can be extended infinitely in any of the two dimensions.
The general form of a plane in 3D is a first-degree equation in x, y, and z, i.e., We represent a plane in 3D as
(ax + by + cz + d = 0)
where:
- (x, y, z) represents the coordinates of a variable point on the plane.
Solid
A solid is a 3-D concept that we also call space. We defined the solid as the extended plane that has three dimensions: length, breadth, and height. A solid can be extended infinitely to incorporate all the space in 3-D.
Vector Form of Equation of a Plane in Normal Form
The vector equation of a plane in normal form is
\pi : \vec{r} \cdot \hat{n} = d
where:
- π represents a plane in 3D space.
({\vec {r}} . {\vec{n}} = d) - The
{\vec{r}} vector is the position vector of a general point lying on the plane. - n̂ is the unit vector normal to the plane
- d is the distance of a plane from the origin
The vector equation of the plane
Example: The vector equation of the plane in 3D space, which is at a distance of 8 units from the origin and normal to the vector (2i + 2k + 2k), is given by?
d = 8 and
{\vec {n}} = (2 i+ j + 2 k)n̂ = (2 i+ j + 2 k) / √(22 + 12 + 22)
n̂ = (2 i+ j + 2 k) / √9
n̂ = (2/3) i+ (1/3) j + (2/3) k
Hence the required vector equation of the plane in normal form is
{\vec {r}} . ((2/3) i+ (1/3) j + (2/3) k) = 8 which can be simplified as{\vec {r}} . (2 i+ 1 j + 2 k) = 24
Cartesian Form of Equation of a Plane in Normal Form
π: lx + my + nz = p
where:
- π again represents a plane in 3D space
- l, m, n are the DCs, i.e., direction cosines of the normal to the plane, always satisfy this condition (l2 + m2 + n2 = 1)
- p is the distance of the plane from the origin
If a, b, and c are not the direction cosines of the normal to the plane, then we have to follow these steps:
Step 1: Keep the terms of x, y, and z on the LHS and take the constant term d on the RHS.
Step 2: If the constant term on the RHS is negative then make it positive by multiplying with (-1) on both sides of the equation.
Step 3: Divide term on the both sides of the equation by √(a2 + b2 + c2).
After applying these steps, the coefficients of x, y, and z on the LHS will become the direction cosines of the normal to the plane, and the constant term on the RHS will become the distance of the plane from the origin.
Example: A plane in the 3D space is represented by (2x + y + 2z - 24 = 0), then the Cartesian equation of this plane in the normal form is given by.
Given equation of plane,
2x + y + 2z = 24
Dividing both sides of the above equation by √(22 + 12 + 22) = √9 = 3
The given equation of plane in cartesian form is,
(2/3) x + (1/3) y + (2/3) z = 8
Here,
l = 2/3 , m = 1/3 , n = 2/3 are the direction cosines
p = 8 is the distance from the origin.
Distance of a Point from a Plane in Cartesian Form
The distance of a point P(xo, yo, zo) from a plane
π:a x + b y + c z +d = 0
in Cartesian form is defined as the length (L) of the perpendicular drawn from that point to the plane. That is calculated using the formula,
L = \frac{\left| a x_0 + b y_0 + c z_0 + d \right|}{\sqrt{a^2 + b^2 + c^2}}
Example: Find the distance of the point (2, 1, 0) from the plane (2 x + y + 2 z + 5 = 0).
xo = 2, yo = 1, zo = 0
a = 2, b = 1, c = 2, d = 5
L = |(2 × 2) + (1 × 1) + (0 × 2) + 5| / √(22, 12, 22)
⇒ L = 10 / √9
⇒ L = 10/3
Thus, the required distance is 10/3 units.
Distance of a Point from a Plane in Vector Form
The distance of a point P having position vector
L =
\frac{| \vec{a} \cdot \vec{n} - d |}{| \vec{n} |}
Example: The distance of a point with position vector (2 i + j + 0k) from the origin
{\vec {a}} = 2 i + j + 0 k
{\vec {n}} = 2 i + j + 2 k|
{\vec {n}} | = √(22 + 12 + 22) = √9 = 3d = 5 (given)
{\vec {a}}.{\vec {n}} = (2 × 2) + (1 × 1) + (0 × 2) = 5⇒ L = |5 - 5| / 3
⇒ L = 0
Thus, the required distance is 0 units.
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Points, Lines, and Planes Solved Examples
Example 1: Find the distance between points (2, 4) and (5, 6).
Given point, (2, 4) and (5, 6)
Comparing with (x1, y1) and (x2, y2)
(x1, y1) = (2, 4) and (x2, y2) = (5, 6)
Using the distance formula,
d = √[(x2 - x1)2 + (y2 - y1)2]
⇒ d = √[(5-2)2 + (6-4)2]
⇒ d = √13
Thus, the distance between two points is √13 units.
Example 2: Find the distance of the point (0, 1, 0) from the plane (3y + 4z = 7).
Given Point,
(xo, yo, zo) = (0, 1, 0)
xo = 0, yo = 1, zo = 0
Equation of the plane,
3 y + 4 z = 7
Comparing with ax + by +cz + d = 0
a = 0, b = 3, c = 4, d = -7
Distance of a point form a plane is given using the formula,
L = |a xo + b yo + c zo + d| / √(a2 + b2 + c2)
⇒ L = |0 + (3 × 1) + (4 × 0) - 7| / √(02, 32, 42)
⇒ L = |3 - 7| / √(25)
⇒ L = 4/5
Thus, the required distance is 4/5 units
Example 3: Find the distance of a point (5, 3, 0) from the plane
{\vec {a}} = 5 i +3 j + 0 k
{\vec {n}} = 4 i +3 j + 0 k|
{\vec {n}} | = √(42+ 32+ 02) = √(25) = 5d = 8 (given)
{\vec {a}} .{\vec {n}} = (5 × 4) + (3 × 3) + (0 × 0) = 29⇒ L = |29 - 8| / 5
⇒ L = 21/5
Thus, the required distance is 21/5 units
Practice Questions on Points, Lines and Planes
Question 1: Identify which of the following sets of points are collinear: A(2, 3), B(5, 7), C(8, 11).
Question 2: Determine if the points D(4, 6), E(4, 6), and F(4, 8) lie on the same line.
Question 3: Find the equation of the line passing through the points P(2, 1) and Q(4, 5) in slope-intercept form.
Question 4: Given three non-collinear points, how many distinct lines can be drawn through them?
Question 5: Determine the distance between the points R(3, 2, 4) and S(7, 5, 8).
Question 6: Determine whether the points G(2, 3, 1), H(4, 1, 3), and I(6, 5, 2) lie on the same plane.