Question

Three points with position vectors `vec(a), vec(b), vec(c ) ` will be collinear if there exist scalars x, y, z such that
A. `x vec(a) + y vec(b)=z vec(c )`
B. `x vec(a) + y vec(b)+z vec(c )=0`
C. `x vec(a) + y vec(b)+z vec(c )=0, " where " x+y+z=0`
D. `x vec(a) + y vec(b)=vec(c ).`

Answer 1

Correct Answer - C
Let, A, B, C be the points with position vectors ` vec(a), vec(b), vec(c ) ` respectively. These points will be collinear, iff
`vec(AB) = lambda vec(AC)`
`rArr vec(b) -vec(a)=lambda (vec(c )-vec(a))`
`rArr (lambda-1)veca+vecb+(-lambda)vecc=vec0`
`rArr x veca+yvecb +z vecc = vec0, " where " x= lambda-1, y=1, z=-lambda`
`rArr x vec a =y vec b + z vec c = vec 0, " where " x+y+z=0`