Tamil Nadu Board of Secondary EducationHSC Arts Class 11th

The position vectors abca→,b→,c→ of three points satisfy the relation abc2a→-7b→+5c→=0→. Are these points collinear? - Mathematics

Sum

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.

Concept: Representation of a Vector and Types of Vectors
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