Three-dimensional geometry, also known as 3D geometry, is a branch of mathematics that deals with the properties and relationships of objects in three-dimensional space. In contrast to the familiar two-dimensional cartesian plane, which is characterised by two perpendicular axes, 3D space has three mutually perpendicular axes, typically labelled x, y, and z. This allows us to describe the position, orientation, and movement of objects in three dimensions using coordinates and vectors. 

While two-dimensional geometry is concerned with the study of objects in two dimensions, such as points, lines, and shapes on a plane. In 3D geometry, we work with points, lines, planes, and other geometric objects that exist in three-dimensional space, which we can visualise as a three-dimensional coordinate system consisting of x, y, and z axes.

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In this maths article we will learn about three dimensional geometry in detail.

What is Three Dimensional Geometry?

Three-dimensional geometry is a branch of mathematics that studies objects in three-dimensional space. This includes points, lines, planes, and shapes like cubes, spheres, and cones. To describe the position of these objects, we use coordinates in the form of ordered triples (x, y, z) of real numbers. With this foundation, we can explore a range of topics in 3D geometry, including vector operations, matrices, transformations like rotations, translations, and projections.

Moreover, we can calculate distances, angles, and areas of various shapes in 3D space, and use this information to solve problems related to geometric optimization, spatial analysis, and design. In terms of formulas, we may encounter concepts like the distance formula, which measures the distance between two points in 3D space, and the dot product formula, which calculates the angle between two vectors. 

Let's look at some important terms used in three-dimensional geometry.

Let us learn the terms used in three-dimensional geometry:

Direction Cosines: If a line forms an angle α, β, γ in the positive direction concerning X-axis, Y-axis and Z-axis, respectively, then cos α, cos β, and cos γ are called its direction cosines. 

The direction cosines are commonly denoted as l, m, and n.i.e. l is equal to cos α, m is equal to cos β and n is equal to cos γ. Whereas the angles α, β, γ are recognized as direction angles.

Some important points on direction cosines are mentioned below

• If l, m, n represent the direction cosines of a line, then
• \(l^2+m^2+n^2=1\)
• Direction cosines of the X-axis are given as 1, 0, 0
• Direction cosines of the Y-axis are given as 0, 1, 0
• Direction cosines of the Z-axis are given as 0, 0, 1
• If a, b and c represent the direction ratios of a line, then:
• \(l=\frac{a}{\sqrt{a^2+b^2+c^2}},\ m=\frac{b}{\sqrt{a^2+b^2+c^2}}and\ \ n=\frac{c}{\sqrt{a^2+b^2+c^2}}\)

Direction Ratios: Three numbers, say a, b, c, proportional to the direction cosines, say l, m, n of a line, are acknowledged as the direction ratios of the line. Thus a, b, and c are termed the direction ratios of a line provided.

\(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}\)

Distance Formula: The distance between two points assumes A (x1, y1, z1) and B (x2, y2, z2) in a three-dimensional space is presented by:

\(AB=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}\)

Section Formula: If A (x1, y1, z1) and C (x2, y2, z2) are two points in a space and let B be a point on the line segment joining A and B such that

• It divides AC internally in the proportion m:n. Then, the coordinates of B are:
• \(\left(\frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n},\ \frac{mz_2+nz_1}{m+n}\right)\)
• It divides AC externally in the ratio m:n (m ≠ n). Then, the coordinates of B are:
• \(\left(\frac{mx_2-nx_1}{m-n},\ \frac{my_2-ny_1}{m-n},\ \frac{mz_2-nz_1}{m-n}\right)\)

Learn about Geometric Shapes

Lines in Three Dimensional Geometry

In 3d geometry, a line is interpreted as a straight one-dimensional figure with no thickness and can extend endlessly in both directions.

A line segment can be defined as a part of a line with determined endpoints. Also, know some important points regarding the lines below.

• The two lines are said to be perpendicular if and only if l1 ⋅ l2 + m1 ⋅ m2 + n1 ⋅ n2 = 0
• The same two lines are said to be parallel if and only if:
• \(\frac{l_1}{l_2}=\frac{m_1}{m_2}=\frac{n_1}{n_2}\)
• The two lines are said to be perpendicular if and only if a1 ⋅ a2 + b1 ⋅ b2 + c1 ⋅ c2 = 0
• The same two lines are said to be parallel if and only if:
• \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)

Equations Regarding Lines

• If a, b, c is the direction ratios of a line crossing through the point (x1, y1, z1), then the equation of the line is presented by:
• \(\left(\frac{\left(x-x_1\right)}{a},\ \frac{\left(y-y_1\right)}{b},\ \frac{\left(z-z_1\right)}{c}\right)\)
• If l, m, n are the direction cosines of a line passing through the point (x1, y1, z1), then the equation of the line is given by:
• \(\left(\frac{\left(x-x_1\right)}{l},\ \frac{\left(y-y_1\right)}{m},\ \frac{\left(z-z_1\right)}{n}\right)\)
• The equation of a line passing through the two points (x1, y1, z1) and (x2, y2, z2) is presented by:
• \(\left(\frac{x-x_1}{x_2-x_1},\ \frac{y-y_1}{y_2-y_1},\ \frac{z-z_1}{z_2-z_1}\right)\)
• Equation of a line passing through the point (x1, y1, z1) and parallel to the line having direction ratios a, b, c is given by:
• \(\left(\frac{\left(x-x_1\right)}{a},\ \frac{\left(y-y_1\right)}{b},\ \frac{\left(z-z_1\right)}{c}\right)\)

Angle Between Two Lines

The angle θ between two lines whose direction cosines are \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) is given by: cos θ = \(l_1 ⋅ l_2 + m_1 ⋅ m_2 + n_1 ⋅ n_2\)

The angle θ between two lines whose direction ratios are proportional to \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) respectively is furnished by:

\(\cos\ \theta=\ \left|\frac{\left(a_1a_2+b_1b_2+c_1c_2\right)}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|\)

Moving towards the next important concept, i.e. planes. We will also cover the equation of a plane in a different form, distance measurement from point and line.

Representation Of A Line in 3D Geometry

The equation of a line in a three-dimensional cartesian system can be found using two main methods. The two methods to write the equation of a line are given below.

The equation of a line that goes through a point 'a' and is in the direction of a vector 'b' is written as:
r = a + λb

The equation of a line that passes through two known points, a and b, can be written as:
r = a + λ(b - a)

Plane in Three Dimensional Geometry

Starting with the equation of a plane in a different form:

• The general equation ax + by + cz + d = 0 represents a plane where a, b and c are constants followed by the condition a, b, c ≠ 0.
• The equation of the plane passing through the origin is given by ax + by + cz = 0.
• The equation of a plane crossing through the point (x1, y1, z1) is given by: \(a\left(x-x_1\right)+b\left(y-y_1\right)+c\left(z-z_1\right)=0\)
• If l, m, n are the direction cosines of the normal to the plane and p is the perpendicular distance of the plane from the origin, then the equation of the plane is furnished by lx + my + nz = p.
• If a, b and c are x, y, and z are the intercept on the corresponding X, Y, and Z-axis, then the equation of the plane is:
• \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\)
• Equation of XY plane– the plane is z = 0.
• Equation of YZ plane– the plane is x = 0.
• Equation of XZ plane– the plane is y = 0.
• The equation of plane parallel to YZ- plane and at a distance p is given by x = p.
• The equation of plane parallel to ZX- plane and at a distance p is given by y = p.
• Equation of plane parallel to XY- plane and at a distance, p is given by z = p.

Equation of a plane passing through the three points A(x1, y1, z1), B (x2, y2, z2), and C (x3, y3, z3) is given by:

\(\left|\begin{matrix}x-x_1&y-y_1&z-z_1\\

x_2-x_1&y_2-y_1&z_2-z_1\\

x_3-x_1&y_3-y_1&z_3-z_1\end{matrix}\right|=0\)

Distance of a Plane from a Point

The perpendicular distance of a plane ax + by + cz + d = 0 from a point P (x1, y1, z1) is given by:

\(\left|\frac{ax_1+by_1+cz_1+d}{\sqrt{a^2+b^2+c^2}}\right|\)

Distance Between Two Parallel Planes

The distance between two parallel planes having equation ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is furnished by:

\(\left|\frac{d_1-d_2}{\sqrt{a^2+b^2+c^2}}\right|\)

Intersection of Two Planes

If a1 x + b1 y + c1 z + d = 0 and a2 x + b2 y + c2 z + d = 0 represents two different planes, then the equation of a plane passing through the intersection of these planes is given by:

=(a1 x + b1 y + c1 z +d) + λ (a2 x + b2 y + c2 z +d) = 0, where λ is a scalar.

Representation Of A Plane in 3D Geometry

The equation of a plane in a cartesian coordinate system can be found in different ways depending on the given information about the plane. Below are the four different ways to write the equation of a plane.

Normal Form: The equation of a plane at a perpendicular distance d from the origin, with a unit normal vector n̂, is
→r · n̂ = d

Perpendicular to a given Line and through a Point: The equation of a plane that is perpendicular to a given vector →N, and passes through a point →a, is
(→r − →a) · →N = 0

Through three Non-Collinear Points: The equation of a plane passing through three non-collinear points →a, →b, and →c, is
(→r − →a) · [(→b − →a) × (→c − →a)] = 0

Intersection of Two Planes: The equation of a plane that passes through the intersection of two planes
→r · n̂₁ = d₁, and →r · n̂₂ = d₂, is
→r · (→n₁ + λ→n₂) = d₁ + λd₂

Coordinate System in Three Dimensional Geometry

In three-dimensional geometry, a point is located in space by its coordinates, usually denoted by \((x, y, z)\). This means that the point is located at a distance of \(x\) units along the \(x\)-axis, \(y\) units along the

\(y\)-axis, and \(z\) units along the \(z\)-axis.

The coordinate system in three-dimensional geometry is a three-axis system, also known as the Cartesian coordinate system. The three axes are labelled \(x\), \(y\), and \(z\), and they are mutually perpendicular to each other. The origin of the coordinate system is the point \((0, 0, 0)\), where all three axes intersect.

For a comprehensive overview of the fundamental concepts, theorems, and formulas of coordinate or analytic geometry, you may consult the lesson on coordinate geometry, which delves deeper into coordinate planes and systems.

Point in Three-Dimensional Geometry

There are two ways to represent a point in three-dimensional geometry: in cartesian form or in vector form. The point can be expressed using either of these two forms of representation.

Cartesian Form: In three-dimensional geometry, a point can be represented in cartesian form using three coordinates, with reference to the \(x\)-axis, \(y\)-axis, and \(z\)-axis. The three coordinates of any point are denoted as \((x, y, z)\), where \(x\) is referred to as the abscissa, \(y\) as the ordinate, and \(z\) as the applicate.

Vector Form: The vector form of representation of a point \(P\) is a position vector \(OP\) and is written as \(\vec{OP}=x\hat{i}+y\hat{j}+z\hat{k}\), where \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) are the unit vectors along the \(x\)-axis, \(y\)-axis, and \(z\)-axis respectively.

Rectangular Coordinate System

A rectangular coordinate system, also known as a Cartesian coordinate system, is a system used to represent points and graph functions in two or three dimensions. It consists of two or three perpendicular lines intersecting at a point called the origin.

The horizontal line is called the \(x\)-axis in two dimensions, and the vertical line is called the \(y\)-axis. 

The origin is denoted by \((0,0)\), where the \(x\) and \(y\) axes intersect. Points in the coordinate system are represented by an ordered pair \((x, y)\), where \(x\) represents the horizontal distance from the origin and \(y\) represents the vertical distance from the origin.

In three dimensions, there is an additional perpendicular line called the \(z\)-axis, which is oriented perpendicular to the \(x\) and \(y\) axes. Points in the coordinate system are represented by an ordered triple \((x, y, z)\), where \(x\), \(y\), and \(z\) represent the distances along the \(x\), \(y\), and \(z\) axes respectively.

The rectangular coordinate system is widely used in various fields of mathematics, science, engineering, and technology for representing and analyzing data, functions, and equations.

Projection in 3D space

Projection in 3D space represents a three-dimensional object or scene onto a two-dimensional surface or plane. This is necessary for many applications, such as computer graphics, engineering, architecture, and art.

A line of sight or projection plane is chosen to perform a projection, and the 3D object or scene is projected onto it. The projection plane can be positioned and oriented in different ways to create different perspectives or views of the object.

Let \(AB\) be a line segment. It’s projection on a line \(PQ\), \(AB\cos(\theta)\), where \(\theta\) is the angle between \(AB\) and \(PQ\) or \(CD\)

Various mathematical techniques and algorithms perform projections in 3D space, such as matrix transformations and ray tracing.

Coplanarity of Two Lines in Three Dimensional Geometry

Lines that are in the same plane are coplanar. These lines are said to be in the same three-dimensional space. The coplanarity of two lines is proved using the condition in vector form and cartesian form.

In a three-dimensional space:-

• Two lines are coplanar if there is a plane that includes them both. This is possible only if the lines are parallel or intersect.
• Three points are always coplanar, and the plane is unique if they are not collinear.
• Four points may or may not lie in the same plane.
• Distance geometry determines whether points are coplanar by determining the distance between them.

Let two lines are \(\frac{x-x_{1}}{l_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}}\) and \(\frac{x-x_{2}}{l_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}}\). Two lines are coplanar if and only if 

\(\begin{vmatrix}x_{2}-x_{1}&y_{2}-y_{1}&z_{2}-z_{1}\\l_{1}&m_{1}&n_{1}\\l_{2}&m_{2}&n_{2}\\\end{vmatrix}\).

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Examples on 3D Geometry 

Example 1: What is the equation of a line in three-dimensional geometry, passing through the points (1, 2, -3), and (4, -1, 0)?

Solution:

The given points are (1, 2, -3), and (4, -1, 0).

The equation of a line passing through two points is:
r = a + λ(b − a)

→r = (1→i + 2→j − 3→k) + λ[(4→i − 1→j + 0→k) − (1→i + 2→j − 3→k)]

→r = (1→i + 2→j − 3→k) + λ(3→i − 3→j + 3→k)

So,
x→i + y→j + z→k = (1 + 3λ)→i + (2 − 3λ)→j + (−3 + 3λ)→k

Now we write in standard form:
(x − 1)→i + (y − 2)→j + (z + 3)→k = 3λ→i − 3λ→j + 3λ→k

Therefore,
(x − 1)/3 = (y − 2)/(−3) = (z + 3)/3

So, the equation of the line passing through the two points is:
(x − 1)/3 = (y − 2)/(−3) = (z + 3)/3

Example 2: What is the equation of a line in three-dimensional geometry, passing through the points (2, -1, 4), and (5, 0, -2)?

Solution:

The given points are (2, -1, 4), and (5, 0, -2).

The equation of a line passing through two points is:
r = a + λ(b − a)

→r = (2→i − 1→j + 4→k) + λ[(5→i + 0→j − 2→k) − (2→i − 1→j + 4→k)]

→r = (2→i − 1→j + 4→k) + λ(3→i + 1→j − 6→k)

So,
x→i + y→j + z→k = (2 + 3λ)→i + (−1 + λ)→j + (4 − 6λ)→k

Now we write in standard form:
(x − 2)→i + (y + 1)→j + (z − 4)→k = 3λ→i + λ→j − 6λ→k

Therefore,
(x − 2)/3 = (y + 1)/1 = (z − 4)/(−6)

So, the equation of the line passing through the two points is:
(x − 2)/3 = (y + 1)/1 = (z − 4)/(−6)

Example 3: If a variable plane forms a tetrahedron of constant volume \(64\) \(K^{3}\) with the coordinate planes then the lows of the centroid of the tetrahedron is \(xyz = uK^{3}\). Find \(u\).

Solution: Let the equation of the plane be \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\).

Centroid of the tetrahedron is \(\left(\frac{a}{4},\frac{b}{4},\frac{c}{4}\right)\).

Volume of the tetrahedron \(=\frac{abc}{6}=64K^{3}\).

So letting 

\(\frac{a}{4} = x\), \(\frac{b}{4} = y\), \(\frac{c}{4} = z\).

We have 

\(\frac{abc}{6}=\frac{4^{3}xyz}{6}=64K^{3}\)

\(\therefore\) \(xyz=6K^{3}\)

On comparing, we have \(u = 6\).

We hope you found this article regarding 3D Geometry was informative and helpful, and please do not hesitate to contact us for any doubts or queries regarding the same. You can also download the Testbook App, which is absolutely free and start preparing for any government competitive examination by taking the mock tests before the examination to boost your preparation. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: