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Question

Given position vectors ¯a,¯b,¯c of points A, B, C for a triangle. The centroid can be given by

A
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B
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C
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D
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Solution

The correct option is A
Let’s construct the triangle BC. Let P, M, N be the mid points of sides AB, BC and CA. By section formula P, M and N can be given by ¯a+¯b2,¯b+¯a2and¯c+¯a2. Apart from that we also know the fact that centroid divides the line joining a vertex and midpoint of opposite side in the ratio 2: 3. ¯G=¯a+23(¯¯¯¯¯¯¯¯¯¯AM) ¯¯¯¯¯¯¯¯¯¯AM=(¯b+¯c2)¯a¯¯¯¯¯¯¯¯¯¯AM=¯b+¯c2¯a2¯G=¯a+23(¯b+¯c2¯a2)=¯a+¯b+¯c3
Centroid can be given by ¯a+¯b+¯c3

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