Vector Algebra Question 271
Question: If $ A,,B,,C $ are the vertices of a triangle whose position vectors are a, b, c and G is the centroid of the $ \Delta ABC, $ then $ \overrightarrow{GA}+\overrightarrow{GB},+\overrightarrow{GC} $ is
[Karnataka CET 2000]
Options:
A) 0
B) $ \overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C} $
C) $ \frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} $
D) $ \frac{\mathbf{a}+\mathbf{b}-\mathbf{c}}{3} $
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Answer:
Correct Answer: A
Solution:
- Position vectors of vertices A, B and C of the triangle ABC = a, b and c. We know that position vector of centroid of the triangle (G) = $ \frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} $ . Therefore , $ \overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC} $ $ =( \mathbf{a}-\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} )+( \mathbf{b}-\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} )+( \mathbf{c}-\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} ) $ $ =\frac{1}{3}[2\mathbf{a}-\mathbf{b}-\mathbf{c}+2\mathbf{b}-\mathbf{a}-\mathbf{c}+2\mathbf{c}-\mathbf{a}-\mathbf{b}]=\mathbf{0} $ .
BETA
