Question

The position vectors `vec"a", vec"b", vec"c"` of three points satisfy the relation `2vec"a" - 7vec"b" + 5vec"c" = vec0`. Are these points collinear?

Answer 1

Let A, B, C b the three points whose position vectors are `vec"a", vec"b"` and `vec"c"`

`vec"OA" = vec"a"`

`vec"OB" = vec"b"`

`vec"OC" = vec"c"`

The position vectors satisfy the condition

`2vec"a" - 7vec"b" + vec"c"` = 0

`vec"a" + 5vec"c" = 7vec"b"`

`2vec"a" + 5vec"c" - 7vec"a" = 7vec"b" - 7vec"a"`

`5vec"c" - 5vec"a" = 7(vec"b" - vec"a")`

`5(vec"c" - vec"a") = 7(vec"b" - vec"a")`

`vec"c" -vec"a" = 7/5(vec"b" - vec"a")`

`vec"c" -vec"a" = lambda(vec"b" - vec"a")`  .......(1)

`vec"AB" = vec"OB" - vec"OA"`

`vec"AB" = vec"b" - vec"a"`

`vec"AC" = vec"OC" - vec"OA"`

`vec"AC" = vec"c" - vec"a"`

(1) ⇒ `vec"AC" = lambda  vec"AB"`

∴ `vec"AC"` and `vec"AB"` are parallel vectors and A is a common point.

∴ Yes, A, B, C are collinear.