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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?
Solution
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.
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