The diagonals of a parallelogram are represented by R → 1 = 3 i ^ - 2 j ^ + 7 k ^ , R → 2 = 5 i ^ + 6 j ^ - 3 k ^ , find the area of the parallelogram.

Given, R → 1 = 3 i ^ - 2 j ^ + 7 k ^ R → 2 = 5 i ^ + 6 j ^ - 3 k ^ Area of the parallelogram represented by the two vectors Is given by A r e a = A × B A r e a = 3 i ^ - 2 j ^ + 7 k ^ × 5 i ^ + 6 j ^ - 3 k ^ A r e a = 36 i ^ + 26 j ^ + 8 k ^ A r e a = 36 2 + 26 2 + 8 2 = 2036 A r e a = 45 . 12 s q . u n i t

The diagonals of a parallelogram are represented by R → 1 = 3 i ^ - 2 j ^ + 7 k ^ , R → 2 = 5 i ^ + 6 j ^ - 3 k ^ , find the area of the parallelogram.

Given, R → 1 = 3 i ^ - 2 j ^ + 7 k ^ R → 2 = 5 i ^ + 6 j ^ - 3 k ^ Area of the parallelogram represented by the two vectors Is given by A r e a = A × B A r e a = 3 i ^ - 2 j ^ + 7 k ^ × 5 i ^ + 6 j ^ - 3 k ^ A r e a = 36 i ^ + 26 j ^ + 8 k ^ A r e a = 36 2 + 26 2 + 8 2 = 2036 A r e a = 45 . 12 s q . u n i t

A particle gets displaced through i ^ + 2 j ^ - 4 k ^ due to a force 2 i ^ - 3 j ^ - 6 k ^ . The displacement and force are measured in S.I. units. Find the work done.

Given, Displacement, S = i ^ + 2 j ^ - 4 k ^ Force, F = 2 i ^ - 3 j ^ - 6 k ^ Work done is given by W = F . S ⇒ W = ( i ^ + 2 j ^ - 4 k ^ ) . ( 2 i ^ - 3 j ^ - 6 k ^ ) ⇒ W = = 2 - 6 + 24 ⇒ W = 20 N

Show that A → · B → = 1 4 | A → + B → | 2 - | A ¯ - B → | 2 .

Taking R . H . S = 1 4 A → + B → 2 - A → - B → 2 = 1 4 A → + B → 2 - 1 4 A → - B → 2 = 1 4 A 2 + B 2 + 2 A B cos θ - 1 4 A 2 + B 2 - 2 A B cos θ = A B cos θ = A → . B → = L . H . S