IIT JEE , AIEEE Forum - Mathematics , Vectors

ramkumar_november

let a = 2i-j+3k

b = 2i+4j-2k

now let c be parallel to b and d be perpendicular to b

we have to express a = c + d

since c is parallel to b . let c = mb = 2m i + 4m j - 2m k

d=a-c

d= 2i - j +3k - 2m i - 4m j +2m k

d= (2-2m)i - (1+4m)j + (3+2m)k

since d is perpendicular to b ....... their dot product is 0

(2-2m)2 -(1+4m)4 -2(3+2m) =0

4 - 4m -4 -16m -6 -4m = 0

-24m = 6

m = -(1/4)

so c = - i/2 - j + k/2

d = 2i - j +3k +i/2 + j -k/2

d = (5/2)i + (5/2)k

thus a = c + d

where c is parallel to b and d is perpendicular to b

clear with my sloution.........????