Vectors and 3D Geometry
Board: JEE Advanced | Exam: JEE Advanced
JEE Advanced study notes for Vectors and 3D Geometry by Ajay Yadav. In-depth coverage of advanced topics and problem-solving techniques.
Key Concepts
Vector Triple Product
a×(b×c)=(a·c)b-(a·b)c. (a×b)×c=(a·c)b-(b·c)a.
Quadruple Products
(a×b)·(c×d)=(a·c)(b·d)-(a·d)(b·c). (a×b)×(c×d)=[a b d]c-[a b c]d.
Reciprocal System
If a,b,c are non-coplanar, reciprocal vectors: a’=(b×c)/[a b c]. a·a’=1, a·b’=0.
Lines and Planes in Space
Skew lines, shortest distance. Equation of plane through intersection of two planes. Angle between line and plane. Projection of line on plane.
Sphere and Cone
Equation of tangent plane to sphere. Right circular cone. Right circular cylinder.
Important Formulas
| a×(b×c) | (a·c)b-(a·b)c |
| (a×b)·(c×d) | (a·c)(b·d)-(a·d)(b·c) |
| Reciprocal | a' = (b×c)/[a b c] |
Solved Examples
Example 1: Simplify a×(b×c)+b×(c×a)+c×(a×b).
Solution: Using vector triple product, each term expands. Sum = 0.
Practice Questions
- Prove (a×b)×(c×d)=[a b d]c-[a b c]d.
- Find volume of tetrahedron with vertices (0,0,0),(1,0,0),(0,1,0),(0,0,1).
- Find the foot of perpendicular from (1,2,3) to line (x-1)/2=(y-2)/3=(z-3)/4.
- Find equation of sphere with center (1,2,3) and tangent to plane 2x+3y+4z=20.