Maths Three Dimensional Geometry Chapter - Maths JEE Main

3-Dimensional geometry involves the mathematics of shapes in 3D space and involves 3 coordinates in the XYZ plane which are x-coordinate, y-coordinate, and z-coordinate. The shapes that occupy space are called 3D shapes. 3D shapes can also be defined as solid shapes having three dimensions length, width, and height. Three-dimensional space is a geometric three-parameter model in which (there are three axes x,y, and z-axes) all known matter exists. These three dimensions are chosen from the term’s length, width, height, depth, and breadth.

In a 3d space, three parameters are required to find the exact location of a point. 3-dimensional geometry plays a major role in JEE exams as a lot of questions are included in the exam. In this article, we will study the basic concepts of geometry involving 3-dimensional coordinate geometry which will help to understand different operations on a point in a 3d plane.

JEE Main Maths Chapters 2025 

Coordinate System in 3D Geometry

In 3-dimensional coordinate geometry, a coordinate system refers to the process of identifying the position or location of a point in the coordinate plane. To understand more about coordinate planes and systems, refer to the coordinate geometry lesson which covers all the basic concepts, theorems, and formulas related to coordinate or analytic geometry.

The Properties of Three-dimensional Space:

• A point is represented by an ordered triple (x, y, and z) that consists of three numbers, an x-coordinate, a y-coordinate, a z-coordinate.

• In the two-dimensional xy-plane, these coordinates indicate the signed distance along the coordinate axes,

• The x-axis, y-axis, and z-axis, respectively, originate from the origin, denoted by O, which has coordinates (0, 0, and 0). 

Rectangular Coordinate System in Space

The coordinate system defines the position of a vector. In the rectangular coordinate in space, we refer to the three-dimensional space. To demonstrate the position of a vector, mark a point as the origin, represented by the point ‘O’. The distance of any vector is now measured from this standard point.

Let ‘O’ be any point in space called origin and X’OX, Y’OY and Z’OZ be three lines perpendicular to each other and these three lines denote the coordinate X, Y, and Z-axis. The planes XY, YZ, and ZX are called the coordinate planes in space.   

Coordinates of a Point in Space

Consider a point P in space. The position of the point P is given by the coordinates (x, y, z) where x, y, z denotes the perpendicular distance from YZ-plane, ZX-plane, and XY-plane respectively. If the vectors i, j, k are assumed to be the unit vectors along OX, OY, OZ respectively, then the position vector of point P is xi + yj + zk or simply (x, y, z).

If ‘O’ is the origin and P is any point with coordinates (x, y, z) from the origin then the distance vector OP by the distance formula is given by OP = $\sqrt{x^{2} + y^{2} + z^{2}}$

Some Key Points to Remember

• The formula for distance between the points P(x1, y1, z1) and Q (x2, y2, z2) is $\sqrt{(x_{1} – x_{2})^{2} + (y_{1} – y_{2})^{2} + (z_{1} – z_{2})^{2} }$ 

• The point dividing the line joining the points P(x1, y1, z1) and Q (x2, y2, z2) in m : n ratio is 

$\frac{(mx_{2} – nx_{1})}{(m + n)}$,  $\frac{(my_{2} – ny_{1})}{(m + n)}$, $\frac{(mz_{2} – nz_{1})}{(m + n)}$ where m + n ≠ 0. 

• The coordinates of the centroid of a triangle having vertices A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is G $\frac{(x_{1} + x_{2} + x_{3})}{3}, \frac{(y_{1} + y_{2} + y_{3})}{3}, \frac{(z_{1} + z_{2} + z_{3})}{3}$. 

Direction Cosines of a Line

The cosines of the angles made by a directed line segment with the coordinate axes are called the direction cosines of that line.

As shown in the figure above, if α, β, and γ are the angles made by the line segment with the coordinate axes then these angles are said as direction angles; and the cosines of these directional angles are the direction cosines of the line. 

Also, cos α, cos β, and cos γ are called the direction cosines and are denoted by l, m, and n respectively.

l = cos α, 

m = cos β and 

n = cos γ

If three numbers are in proportion with the direction cosines of a line, then they are called the direction ratios. Hence, if ‘a’, ‘b’, and ‘c’ are the direction ratios, and l, m, n are the direction cosines then, we must have,

$\frac{a}{l} = \frac{b}{m} = \frac{c}{n}$.

Some Key Points of Direction Cosines

• As we know that l = cos α, m = cos β, and n = cos γ also -1< cos x< 1 ∀ x ∈ R, so l, m, and n are real numbers with values varying between -1 to 1. So, direction cosine ∈ −1,1−1,1.

• The angles between the x-axis and the coordinate axes are 0°, 90°, and 90°. So the direction cosines are cos 0°, cos 90° and cos 90° i.e. 1, 0, 0 respectively.

• The direction cosine of the x, y, and z axes are (1,0,0), (0,1,0) and (0,0,1).

• The direction cosines of a line parallel to any coordinate axis are equal to the direction cosines of the corresponding axis.

• The direction cosines are associated by the relation l2 + m2 + n2 =1.

• If the given line is inverted, then the dc’s will be cos (π − α), cos (π − β), cos (π − γ) or − cos α, − cos β, − cos γ.

• Thus, a line can have two sets of direction cosines according to its direction.

• If the two lines are parallel, then their direction cosines are always the same.  

• Direction ratios and direction cosines are proportional to each other and hence for a given line, there can be infinitely many direction ratios.

Perpendicular Distance of a Point From a Line

Let us assume AB be the straight line passing through the point A (a, b, c) and having direction cosines l, m, and n. Now, if AN is the projection of line AP on the straight-line AB then we have,

AN = l(x – a) + m(y – b) + n(z – c),

and  $\begin{equation}AP = \sqrt{(x - a)^2 + (y - b)^2 + (z - c)^2}\end{equation}$

∴ Perpendicular distance of point 

$PN = \sqrt{AP^2 + AN^2}$

(Image will be Uploaded soon)

The length of the perpendicular from a point (x1, y1, z1) to a plane ax+ by + cz + d = 0 is

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

Equation of Plane in Different Forms

• General equation of a plane is $ax + by + cz + d = 0$

• Equation of the plane in Normal form is $lx + my + nz = p$ where p is the length of the normal from the origin to the plane and (l, m, n) be the direction cosines of the normal.

• The equation to the plane passing through $P(x_1, y_1, z_1)$ and having direction ratios

• (a, b, c) for its normal is a(x – x1) + b(y – y1) + c (z – z1) = 0

• The equation of the plane passing through three non-collinear points (x1, y1, z1),

• $(x_2, y_2, z_2)$ and $(x_3, y_3 , z_3)$ is $\begin{vmatrix}(x-x_1)&(y-y_1)&(z-z_1) \\ (x_2-x_1)&(y_2-y_1)&(z_2-z_1) \\ (x_3-x_2)&(y_3-y_2)&(z_3-z_2) \end{vmatrix}$

• The equation of a plane that meets the x-axis at ‘a,’ the y-axis at ‘b,’ and the z-axis at ‘c’ (a,b, and c are not zero) is given by: x/a + y/b + z/c = 1 (a b c ≠ 0)

What is a Straight Line?

In three dimensional geometry, a straight line is defined as the intersection of two planes. So general equation of straight line:

$a_1x + b_1y + c_1z + d_1 = 0, a_2x + b_2y + c_2z + d_2 = 0$

Different Equation of a Straight Line

Symmetrical Form: Equation of straight line passing through point P (x1, y1, z1) and whose direction cosines are l, m, n is: (x – x1)/l = (y – y1)/m = (z – z1) / n. It's like giving directions to someone to find that line.

Equation of Line Passing Through Two Points: x–x1 / x2– x1 = y–y1 / y2 – y1 = z – y1 / z2 – z1

Section Formula: If P(x, y) divides the line joining A(x1, y1) and B(x2, y2) in the ratio of m:n is: x = (mx2 + nx1)/ (m+n) and y = (my2 + ny1)/ (m+n).

Intercept Form: If a straight line makes an intercept of say ‘a’ and ‘b’ on x and y axis respectively, then: x/a + y/b = 1

Some Important Facts

For x-axis, we have l = 1 and m = n = 0. Hence,

Projection on x- axis $= 1(x_2-x_1) + 0(y_2-y_1) + 0(z_2-z_1) = (x_2-x_1)$

Projection on y- axis $= 0(x_2-x_1) + 1(y_2-y_1) + 0(z_2-z_1) = (y_2-y_1)$

Projection on z- axis $= 0(x_2-x_1) + 0(y_2-y_1) + 1(z_2-z_1) = (z_2-z_1)$

When you project a line segment onto the coordinate axes (like the x-axis, y-axis, and z-axis), you can find the length of that line segment is: $= \sqrt{a^2 + b^2 + c^2}$

The directions cosines of the line segment along each axis are given by: $± a/√(a2 + b2 + c2), ± b/√(a2 + b2 + c2), ± c/√(a2 + b2 + c2)$

Spheres in Three-Dimensional Geometry

Spheres are essential objects in three-dimensional geometry and play a significant role in mathematics, physics, and engineering. Understanding the properties and equations related to spheres is crucial for success in the Joint Entrance Examination (JEE) Main, especially in the chapter on Three-Dimensional Geometry.

Basic Concepts:

A sphere is a three-dimensional object that is perfectly round, like a ball. It is defined as the set of all points in space that are at a fixed distance (called the radius) from a given point (called the center). In mathematical terms, a sphere with center (h, k, l) and radius 'r' can be represented as:

Equation of a Sphere:

(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2

Where:

• (h, k, l) are the coordinates of the center of the sphere.

• 'r' is the radius of the sphere.

Important Properties and Formulas:

• Surface Area of a Sphere: The surface area of a sphere with radius 'r' is given by the formula = 4πr^2

• Volume of a Sphere: The volume of a sphere with radius 'r' is given by the formula: 

Volume = (4/3)πr^3

• Diameter of a Sphere: The diameter of a sphere is twice the radius, so D = 2r.

• Distance between Two Points and the Sphere: To find the distance 'd' between a point (x₁, y₁, z₁) and the center (h, k, l) of a sphere, you can use the distance formula:

d = √[(x₁ - h)² + (y₁ - k)² + (z₁ - l)²]

• Intersection of a Line and a Sphere: To find the points where a line intersects a sphere, you can substitute the coordinates of the line into the equation of the sphere and solve for 't':

(x - x₁)² + (y - y₁)² + (z - z₁)² = r²

• Equation of a Tangent Plane to a Sphere: The equation of a tangent plane to a sphere with center (h, k, l) and radius 'r' at a point (x₁, y₁, z₁) on the sphere is given by:

(x - x₁)(x - h) + (y - y₁)(y - k) + (z - z₁)(z - l) = r²

• Chord of a Sphere: A chord is a line segment that joins two points on the sphere. The length of a chord can be found using the distance formula mentioned earlier.

Difference Between 2D Shape and 3D Geometric Shape

The following are the differences between a 2D and a 3D form:

• 3D shapes have length, breadth, and height, but 2D shapes only have length and width.

• Two-dimensional shapes have an area but no volume, whereas three-dimensional shapes have both a surface area and a volume.

• Square, triangle, and rectangle are examples of 2D shapes, whereas cube, cuboid, and prism are examples of 3D shapes.

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