3D Geometry Formulas for JEE Mains
Introduction
3D Geometry is an essential topic in Mathematics for JEE Mains aspirants. It deals with the study of geometric shapes in three-dimensional space. In this article, we will provide you with the essential formulas and concepts required to excel in 3D Geometry for JEE Mains.
Basic Concepts
Coordinate Geometry in 3D Space
In 3D space, each point is represented by an ordered triplet (x, y, z) called the coordinates of the point.
- Distance Formula: The distance between two points P(x1, y1, z1) and Q(x2, y2, z2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Direction Cosines and Ratios
Direction cosines and ratios are used to find the direction of a line or a vector in 3D space.
- Direction Cosines: The direction cosines of a line making angles α, β, and γ with the x, y, and z-axes respectively are given by:
l = cos(α), m = cos(β), n = cos(γ)
- Direction Ratios: The direction ratios of a line are given by:
a : b : c = l : m : n
Vector Algebra
Vector algebra is used to solve problems involving vectors in 3D space.
- ** Dot Product**: The dot product of two vectors a and b is given by:
a . b = |a| |b| cos(θ)
- Cross Product: The cross product of two vectors a and b is given by:
a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
3D Geometry Formulas for JEE Mains
Distance and Section Formulae
- Distance between two points: d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
- Section Formula: The coordinates of a point P dividing the line segment joining A(x1, y1, z1) and B(x2, y2, z2) in the ratio m:n are given by:
(mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))
Intersection of Lines and Planes
- Intersection of two lines: The point of intersection of two lines can be found by solving the system of linear equations.
- Intersection of a line and a plane: The point of intersection of a line and a plane can be found by solving the system of linear equations.
Planes and Angles
- Equation of a plane: The equation of a plane is given by:
ax + by + cz + d = 0
- Angle between two planes: The angle between two planes is given by:
cos(θ) = |a1a2 + b1b2 + c1c2| / (√(a1^2 + b1^2 + c1^2) * √(a2^2 + b2^2 + c2^2))
Spherical Coordinates
- Spherical to Cartesian coordinates:
x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), z = r cos(θ)
Conclusion
In this article, we have provided you with the essential formulas and concepts required to excel in 3D Geometry for JEE Mains. These formulas will help you solve problems involving coordinate geometry, direction cosines and ratios, vector algebra, distance and section formulae, intersection of lines and planes, planes and angles, and spherical coordinates. Practice these formulas to improve your skills and score well in JEE Mains.
