Statement -1 : If `veca and vecb` are non- collinear vectors, then points having position vectors `x_(1) vec(a) + y_(1) vec(b) , x_(2)vec(a)+ y_(2) vec(b) and x_(3) veca + y_(3) vecb` are collinear if
`|(x_(1),x_(2),x_(3)),(y_(1),y_(2),y_(3)),(1,1,1)|=0`
Statement -2: Three points with position vectors `veca, vecb , vec c` are collinear iff there exist scalars x, y, z not all zero such that `x vec a + y vec b + z vec c = vec 0, " where " x+y+z=0.`

Options

A. Statement - 1 is True, Statement - 2 is True , Statement - 2 is a correct explanation for Statement - 1.
B. Statement -1 is True, Statement - 2 is True, Statement -2 is not a correct explanation for Statement - 1.
C. Statement - 1 is True, Statement - 2 is False.
D. Statement - 1 is False, Statement - 2 is True.

Answer

A

Explanation

Statement -2 si true
Using statement -2, points `x_(1) veca + y_(1) vec b , x_(2) veca + y_(2) vec b and x_(3) vec a + y_(3) vecb` will be collinear iff there exist scalars l, m, n such that
`l (x_(1)veca +y_(1)vecb) + m(x_(2) veca + y_(2)vec b)+ n( x_(3) vec a + y_(3) vec b) = vec 0,` where `l + m+ n =0`
`rArr (lx_(1)+mx_(2)+nx_(3))vec a + (ly_(1)+my_(2)+ny_(3)) vec b = vec 0`
`rArr lx_(1)+mx_(2)+nx_(3) =0 and ly_(1)+my_(2) + ny_(3) =0`
` " " [ because vec a , vec b " are non-collinear " ]`
Thus, we have,
`lx_(1)+mx_(2)+nx_(3)=0`
`ly_(1)+my_(2) + ny_(3) =0`
`l+m+n=0`
This is a homogeneous system of equations having non-trivial solutions (as l, m, n are not all zero).
`therefore |(x_(1),x_(2),x_(3)),(y_(1),y_(2),y_(3)),(1,1,1)|=0`
So statement -1 is true and statement -2 is a correct explanation for statement -1.