Class 12 Maths - GUJARAT
Three Dimensional Geometry
The Three Dimensional Geometry chapter in Class 12 Gujarat (GSEB) Mathematics builds upon your foundational knowledge of 2D coordinate geometry by introducing the third dimension. You will learn to determine the position of a point in space using direction cosines and direction ratios. The chapter extensively covers the vector and Cartesian equations of lines and planes, the shortest distance between two skew lines, and the angle between lines and planes. Scoring well in this chapter is crucial for board exams as it features straightforward, high-weightage numerical problems that test your visualization and algebraic skills.
Start Learning FreeKey Concepts
Direction Cosines and Direction Ratios
Direction cosines (l, m, n) are the cosines of the angles made by a directed line with the positive axes, where l squared plus m squared plus n squared equals one. Direction ratios (a, b, c) are numbers proportional to these direction cosines.
Equation of a Line
A line in 3D space can be determined if it passes through a given point with position vector a and is parallel to a given vector b. Its vector equation is r = a + lambda b.
Angle Between Two Lines
The acute angle between two lines with direction ratios a1, b1, c1 and a2, b2, c2 is given by the dot product of their direction vectors, calculated using the modulus of (a1a2 + b1b2 + c1c2) divided by the product of their magnitudes.
Shortest Distance Between Two Lines
For skew lines (lines that are neither parallel nor intersecting), the shortest distance is the perpendicular line segment connecting them, calculated using vector cross and dot products.
Equation of a Plane
A plane can be represented in various forms, including normal form (r dot n equals d), passing through a given point perpendicular to a given vector, and passing through three non-collinear points.
Important Formulas
Board Exam Info
In the Gujarat (GSEB) Class 12 Mathematics board exam, Three Dimensional Geometry typically carries around 6 to 8 marks. Questions usually include 1-mark objective questions, 2-mark short questions on direction cosines or angles, and 4-mark long-answer problems focusing on finding the shortest distance between skew lines or equations of planes.
Frequently Asked Questions
Direction cosines are specific bounded values (cos alpha, cos beta, cos gamma) whose sum of squares equals 1, whereas direction ratios are any set of numbers proportional to the direction cosines and can take any real values.
How do I identify whether two lines are intersecting or skew?
First check if the lines are parallel by comparing their direction ratios. If they are not parallel, equate their Cartesian coordinates to find values for parameters like lambda and mu; if a consistent solution exists, they intersect, otherwise they are skew lines.
Are vector forms or Cartesian forms preferred in board exams?
Both forms are equally important in GSEB exams. You should be comfortable converting a line or plane equation from Cartesian form to vector form and vice versa, as questions may specifically ask for one or the other.
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