Important Three Dimensional Geometry Formulas For JEE Maths
The significance of three-dimensional geometry is substantial, with numerous questions featuring in the JEE exam. Presented here is a comprehensive compilation of three-dimensional geometry formulas, aiding students in swift revision prior to the examination.
Three-dimensional Geometry Formulas
1. Vector representation of a point: Position vector of a point P(x, y, z) is
2. Distance formula:
Distance between two points P(x1, y1, z1) and Q(x2, y2, z2) is
3. Distance of P from coordinate axes:
4. Section Formula:
Midpoint:
5. Centroid of a triangle:
6. Incentre of triangle ABC:
7. Centroid of a tetrahedron:
8. Direction cosines and direction ratios:
(i) Direction cosines: let α, β, γ be the angles which a directed line makes with the positive directions of the axes of x, y and z respectively, then cos α, cos β and cos γ are called the direction cosines of the line. The direction cosines are usually denoted by (l, m, n).
Therefore, l = cos α, m = cos β, n = cos γ.
(ii) l2+m2+n2 = 1
(iii) If a, b, c are the direction ratios of any line L then
will be a vector parallel to the line L.
(iv) If l, m, and n are the direction cosines of any line L, then
is a unit vector parallel to the line L.
(v) If l, m, n be the direction cosines and a, b, c be the direction ratios of a vector, then
(vi) If OP = r, the direction cosines of OP are l, m, n then the coordinates of P are (lr, mr, nr).
If the direction cosines of the line AB are l, m, n, |AB| = r and the coordinates of A is (x1, y1, z1) then the coordinates of B are given as (x1 + rl, y1+ rm, z1 + rn).
(vii) If the coordinates P and Q are (x1, y1, z1) and (x2, y2, z2) then the direction ratios of line PQ are a = x2 – x1 , b = y2 – y1 and c = z2 – z1 and the direction cosines of line PQ are:
(vii) Direction cosines of the x-axis is (1, 0, 0).
Direction cosines of the y-axis is (0, 1, 0).
Direction cosines of the z-axis is (0, 0, 1).
9. Angle between two line segments:
If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the acute angle between them, then
The line will be perpendicular if a1a2 + b1b2 + c1c2 = 0, and parallel if a1/a2 = b1/b2 = c1/c2.
10. Projection of a line segment on a line:
If P(x1, y1, z1) and Q(x2, y2, z2) then the projection of PQ on a line having direction cosines l, m, n is |l(x2 – x1) + m(y2 – y1) + n(z2 – z1)|
11. Equation of a plane: General form: ax + by + cz + d = 0, where a, b, c are not all zero, a, b, c, d ∈ R.
(i) Normal form: lx + my + nz = p
(ii) Plane through the point (x1, y1, z1): a(x – x1) + b(y – y1) + c(z – z1) = 0
(iii) Intercept form:
(iv) vector form:
(v) Planes parallel to the axes :
(a) plane parallel to X-axis is by + cz + d = 0
(b) plane parallel to Y-axis is ax + cz + d = 0
(c) plane parallel to Z-axis is ax + by + d = 0
(vi) Plane through origin: Equation of the plane passing through the origin is ax + by + cz = 0.
(vii) Transformation of the equation of a plane to the normal form: ax+by+cz-d = 0 in normal form is
(viii) Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz + λ = 0.
Distance between ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is
(ix) A plane ax + by + cz + d = 0 divides the line segment joining (x1, y1, z1) and (x2, y2, z2) in the ratio
(x) Coplanarity of four points: The points A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) are coplanar if
12. A point and a plane:
(i) distance of the point (x’, y’, z’) from the plane ax+by+cz+d = 0 is given by
Understanding Three-Dimensional Geometry:
Three-dimensional geometry deals with the study of objects in three-dimensional space. Unlike two-dimensional shapes, three-dimensional figures possess length, width, and height, adding an extra layer of complexity to calculations. In the context of JEE and other competitive exams, questions related to vectors, lines, planes, and various spatial relationships are common.
The Significance in JEE Exams:
JEE exams, known for their challenging and diverse question patterns, often include a significant number of questions from three-dimensional geometry. These questions assess your ability to visualize and manipulate objects in three-dimensional space, which is a valuable skill in engineering and related fields. Mastering the relevant formulas can help you save time during the exam and ensure accurate problem-solving.
A Comprehensive Compilation of Formulas:
To excel in three-dimensional geometry, it's essential to have a firm grasp of the formulas governing various concepts. Here's a breakdown of some essential formulas that will aid in your JEE preparation:
Distance Formula:
The distance between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is given by:
- Distance AB = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²
Section Formula: The coordinates of a point P that divides the line joining A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in the ratio m:n are:
x = (mx₂ + nx₁) / (m + n)
y = (my₂ + ny₁) / (m + n)
- z = (mz₂ + nz₁) / (m + n)
Vector Formula: The position vector of a point P(x, y, z) is given by:
- OP = xi + yj + zk
Equation of a Plane: The equation of a plane passing through point P(x₁, y₁, z₁) with normal vector N(a, b, c) is:
- ax + by + cz = ax₁ + by₁ + cz₁
Direction Cosines and Ratios: For a line with direction cosines l, m, and n:
l = cos(α), m = cos(β), n = cos(γ)
- Direction ratios are proportional to the direction cosines.
Angle between Two Lines: The angle θ between two lines with direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂) is given by:
- cos(θ) = |l₁l₂ + m₁m₂ + n₁n₂|
Preparing for Success:
As you prepare for your JEE exams, dedicating time to master three-dimensional geometry formulas is a strategic move. Understanding these formulas not only empowers you to solve problems efficiently but also enhances your overall problem-solving skills. Practice a wide range of questions to solidify your understanding and apply these formulas to real-world scenarios.
In Conclusion:
Three-dimensional geometry is a foundational topic with diverse applications. Its formulas play a pivotal role in solving complex problems encountered in JEE exams and beyond. By internalizing these formulas and practicing diligently, you equip yourself with the tools needed to excel in three-dimensional geometry and set yourself on the path to success in competitive exams and your future endeavors.
Important Three Dimensional Geometry Formulas For JEE Maths FAQs
Q1: What is the formula for 3-dimensional geometry?
Ans: The formulas in 3-dimensional geometry define relationships between points, lines, and planes in a three-dimensional space. Key formulas include distance formula, section formula, vector formulas, and equations for planes.
Q2: Is three-dimensional geometry tough?
Ans: Three-dimensional geometry can be challenging for some due to its spatial complexity. However, with practice and a clear understanding of the concepts, it becomes manageable.
Q3: What is the coordinate geometry formula for IIT JEE?
Ans: One important formula in coordinate geometry for IIT JEE is the distance formula between two points (x₁, y₁, z₁) and (x₂, y₂, z₂): √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)².
Q4: What is the formula for a 3D vector Class 12?
Ans: In Class 12, a common formula for a three-dimensional vector is the position vector of a point P(x, y, z), which is represented as OP = xi + yj + zk, where i, j, and k are unit vectors along the x, y, and z axes respectively.
