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Maths Three Dimensional Geometry Chapter - Maths JEE Main

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Last updated date: 17th Apr 2024
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Exploring Three Dimensional Geometry in 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 2024 


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 = x2+y2+z2

 

Distance from the Origin:

When we talk about the ‘distance from the origin,’ we mean finding how far a point is from the point (0, 0, 0), which is the center or starting point of the three-dimensional coordinate system. This distance is calculated using the principles of three-dimensional space.


The distance from the origin in three-dimensional geometry is calculated using the following formula:


Distance = $\sqrt{x^2 + y^2 + z^2}$


Distance Between 2 Points

In three-dimensional geometry, finding the distance between two points is similar to finding the distance between two points in two-dimensional space (like on a coordinate plane), but it involves three coordinates (x, y, z) instead of just two. Here's how you can calculate the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in 3D space:


Distance = $\sqrt{[(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2]}$

where $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are the coordinates of the two points.


Division of a Line Joining 2 Points

When you have a line joining two points, you can divide it into different segments or parts. This division can be done in various ways depending on what you want to achieve or calculate. It's a common concept in geometry and can be used in problem-solving and analysis.


\[P(x, y, z) = \left(\dfrac{{mx_2 + nx_1}}{{m + n}}, \dfrac{{my_2 + ny_1}}{{m + n}}, \dfrac{{mz_2 + nz_1}}{{m + n}}\right)\]

Here, $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are the coordinates of points A and B, respectively. The values of m and n determine the ratio at which the line segment AB is divided, and (x, y, z) are the coordinates of point P.


Direction Cosines and Direction Ratios of a Line in the Cartesian Plane:

In simple terms, when a line makes angles with the positive x, y, and z axes, we can find some numbers to describe how it's pointing. These numbers are called direction cosines.

$l^2+m^2+n^2=1$


So, if we call the angles α, β, and γ, then we can use cos α, cos β, and cos γ to represent these direction cosines, and we usually call them l, m, and n.


I.e, $\dfrac{l}{a}=\dfrac{m}{b}=\dfrac{n}{c}=\dfrac{\sqrt{l^2+m^2+n^2}}{\sqrt{a^2+b^2+c^2}}=\dfrac{1}{\sqrt{a^2+b^2+c^2}}$


Direction Cosine of Line Joining Two Given Points:

If we have two points, P and Q, with coordinates $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, then we can say it is a direction cosines.


$(l, m, n) = \dfrac{(x_2 - x_1, y_2 - y_1, z_2 - z_1)}{|(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2|$


Angle Between Two Lines

When you have two straight lines, AB and AC, and you want to find the angle θ between them, you can use their direction cosines (numbers that tell us the direction of the lines). To find θ, you can calculate it using this formula: 

$\theta = \cos^{-1}(l_1l_2 + m_1 m_2 + n_1 n_2)$


Now, if you know the direction ratios (numbers that describe how the lines move in space) of the two lines as $a_1, b_1, c_1$, and $a_2, b_2, c_2$, you can find the angle between them using this formula:


Angle $= \cos^{-1}\dfrac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{(a_1^2 + b_1^2 + c_1^2)}\sqrt{(a_2^2 + b_2^2 + c_2^2)}}$


Some Key Points to Remember

  • The formula for distance between the points P(x1, y1, z1) and Q (x2, y2, z2) is (x1x2)2+(y1y2)2+(z1z2)2 

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

(mx2nx1)(m+n)(my2ny1)(m+n), (mz2nz1)(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 (x1+x2+x3)3,(y1+y2+y3)3,(z1+z2+z3)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,

 

al=bm=cn.

 

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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FAQs on Maths Three Dimensional Geometry Chapter - Maths JEE Main

1. What is the definition of 3D Shapes?

3D forms are solid shapes or structures with three dimensions: length, breadth, and height, as opposed to two-dimensional objects with only two dimensions., that is length and width. Faces, edges, and vertices are other significant concepts connected with 3D geometric forms. They have depth, therefore they take up a lot of space. Some 3D shapes have the 2D base and top parts, as well as cross-sections. A cube, for example, has square faces on all of its faces. There are various classifications for 3D shapes. Some have curved surfaces, while others are shaped like pyramids or prisms. These forms are known by the designations cube, cuboid, cylinder, cone, and sphere. Edges, faces, vertices, curved surfaces, lateral surfaces, and volume are all qualities that characterise 3D forms.

2. What are Polyhedrons?

A polyhedron is a three-dimensional form containing polygonal faces such as triangles, squares, and hexagons, as well as straight edges and vertices. A platonic solid is another name for it. Polyhedra are solids with straight sides that have the following properties:

  • Polyhedrons should have straight edges.

  • It must have flat sides known as faces.

  • It must have the corners, called vertices


Polyhedrons, like polygons in two dimensions, are classed into regular and irregular polyhedrons, as well as convex and concave polyhedrons. The following are some of the most common polyhedra:

  • Cube: It includes 6 square faces, 8 vertices, and 12 edges.

  • Cuboid: It includes 8 vertices, 6 rectangular faces and 12 edges.

  • Pyramid: It consists of a single vertex, polygonal base, flat faces and straight edges.

  • Prism: It consists of identical polygon ends and flat parallelogram sides.


Some other examples of regular polyhedrons are octahedrons, tetrahedrons, icosahedrons and dodecahedrons. These regular polyhedrons are also known as platonic solids since their faces are similar to one another.

3. What is a 3D Geometric Shape?

A three-dimensional form or solid is referred to as a 3D shape. They each have a face, an edge, and a vertex. The volume of these forms is determined by the space they occupy. The area exists in 2D forms, whereas surface area exists in 3D shapes. The area of all the faces of a 3D form is referred to as the surface area. Cube, cuboid, cone, and cylinder are examples of 3D forms. Many real-world things that approximate 3D shapes may be found all around us. Real-life examples of 3D shapes include a book, a Coca-Cola can, and a party hat.

4. What are the common properties of 3D Geometric Shape?

The following are some of the most common characteristics of 3D shapes:

The dimensions of 3D forms include length, breadth, and height. A sphere is unique in that it lacks these three dimensions but extends in all three directions.


Faces, vertices, edges, and curved surfaces may or may not exist in 3D forms.


Most 3D forms have polygonal faces, such as triangles, squares, and rectangles.


A three-dimensional object with six square faces, eight vertices, and twelve edges. All of the edges and faces are the same.


By definition, three-dimensional forms have an inside and an outer, separated by a surface. All physical objects that we come into contact with are three-dimensional.

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