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Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero. - Mathematics

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Sum

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

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Solution

Let `vec"a", vec"b" and vec"c"` are the position vectors of the vertices A, B and C respectively.

Then we know that the position vector of the centroid O of the triangle is `(vec"a" + vec"b" + vec"c")/3`

Therefore sum of the three vectors `vec"OA", vec"OB" and vec"OC",` is

`vec"OA" +  vec"OB" + vec"OC" = vec"a" - ((vec"a" + vec"b" + vec"c")/3) + vec"b" - ((vec"a" + vec"b" + vec"c")/3) + vec"c" - ((vec"a" + vec"b" + vec"c")/3)`

`= (vec"a" + vec"b" + vec"c") - 3((vec"a" + vec"b" + vec"c")/3)`

`= vec0`

Hence, Sum of the three vectors determined by the medians of a triangle directed from the vertices is zero. 

Concept: Addition of Vectors
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 23 Algebra of Vectors
Exercise 23.4 | Q 2 | Page 36

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