Vector Algebra
Board: JEE Main |Exam:JEEMain
JEEMainstudynotesforVectorAlgebrabyAjayYadav.
KeyConcepts
BasicOperations
Addition,scalarmultiplication.Vectorjoiningtwopoints.Sectionformula.
DotandCross
a·b=|a||b|cosθ. a×b=|a||b|sinθ n̂. Properties and applications.
Triple Products
Scalar triple: [a b c]=a·(b×c). Volume of parallelepiped. Coplanarity. Vector triple: a×(b×c)=(a·c)b-(a·b)c.
Application
Area of triangle = ½|a×b|. Area of parallelogram = |a×b|. Volume of tetrahedron = (1/6)|[a b c]|.
Important Formulas
| a·b | a1b1+a2b2+a₃b₃ |
| |a×b| | Area of parallelogram |
| [a b c] | Volume of parallelepiped |
Solved Examples
Example 1: Find area of triangle with vertices (1,0,0),(0,1,0),(0,0,1).
Solution: AB=-i+j, AC=-i+k. AB×AC = i+j+k. |AB×AC|/2 = √3/2.
Practice Questions
- Find a·b if |a|=3,|b|=4, angle=60°.
- Find a×b for a=i+2j-k, b=2i-j+3k.
- Find volume of parallelepiped with sides i+j, j+k, k+i.
- Show [a+b b+c c+a] = 2[a b c].
- Find projection of a=i+j+k on b=2i+3j+4k.