How to Solve Three Dimensional Geometry Problems Easily
FAQs on Three Dimensional Geometry Practice Paper
1. What is three-dimensional geometry?
Three-dimensional geometry, also known as 3D geometry, is the study of shapes, lines, and figures in space having three dimensions: length, width, and height. Key concepts include coordinates, distance between points, direction cosines, equations of lines and planes, and angles between them. Main topics include:
- Cartesian coordinate systems
- Equations of lines and planes
- Shortest distance between lines
- Angle between two planes or lines
2. What are direction cosines and direction ratios?
Direction cosines are the cosines of angles that a directed line makes with the positive directions of the coordinate axes, and direction ratios are any three numbers proportional to the direction cosines of a line. For a line with direction ratios (a, b, c), the direction cosines (l, m, n) are calculated as:
- l = a/√(a² + b² + c²)
- m = b/√(a² + b² + c²)
- n = c/√(a² + b² + c²)
3. How do you find the equation of a plane in 3D geometry?
The equation of a plane in three-dimensional geometry can be written as Ax + By + Cz + D = 0, where A, B, C are the direction ratios of the normal to the plane. You can find a plane’s equation using:
- A point through which the plane passes
- A normal vector (perpendicular to the plane)
- By plugging coordinates of a point and the normal vector into the general equation
4. How do you find the distance between two points in 3D space?
To find the distance between two points P(x1, y1, z1) and Q(x2, y2, z2) in 3D geometry:
- Use the distance formula: √[(x2-x1)² + (y2-y1)² + (z2-z1)²]
5. What is the shortest distance between two skew lines in 3D geometry?
The shortest distance between two skew lines (non-parallel, non-intersecting lines) is the length of the perpendicular segment joining them. Steps to find it:
- Write both lines in vector form.
- Find the cross product of their direction vectors.
- Construct a vector between any point on one line to any point on the other.
- Project this vector onto the cross product vector and divide by its magnitude.
6. How do you find the angle between two planes in 3D space?
The angle between two planes is given by the angle between their normal vectors. To calculate it:
- If planes have normals (A1, B1, C1) and (A2, B2, C2), use:
cosθ = (A1A2 + B1B2 + C1C2)/ (√(A1²+B1²+C1²) × √(A2²+B2²+C2²))
7. What is the Cartesian equation of a line in three-dimensional geometry?
The Cartesian equation of a line passing through point (x1, y1, z1) and parallel to direction ratios a, b, c is:
- (x - x1)/a = (y - y1)/b = (z - z1)/c
8. What is the equation of a plane passing through three given points?
To find the equation of a plane passing through three points A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3):
- Form two vectors: AB and AC.
- Find their cross product to get the normal vector.
- Use the point-normal form: (r - a) · n = 0
9. What is the locus of a point equidistant from two given points in 3D?
The locus of a point P(x, y, z) equidistant from two given points A(x1, y1, z1) and B(x2, y2, z2) is a plane and its equation is:
(x - x1)² + (y - y1)² + (z - z1)² = (x - x2)² + (y - y2)² + (z - z2)²
Simplifying this gives the equation of the required plane.
10. State the vector equation of a line passing through two given points.
The vector equation of a line passing through points 𝐀 and 𝐁 is given by:
r = a + λ(b – a), where ‘a’ and ‘b’ are the position vectors of the points and λ is a parameter. This form is fundamental in vector and three-dimensional geometry for expressing the equation of a line.
11. What are the applications of three-dimensional geometry in real life?
Three-dimensional geometry is widely used to solve practical problems in various fields. Main applications include:
- Engineering: Designing structures and mechanical components
- Architecture: Planning buildings and landscapes
- Computer Graphics: 3D modeling and simulations
- Navigation: Air and sea travel coordination
