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JEE Main & Advanced Mathematics Vector Algebra Question Bank Modulus of vector Algebra of vectors

  • question_answer
    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]

    A) 0

    B) \[\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}\]

    C) \[\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3}\]         

    D) \[\frac{\mathbf{a}+\mathbf{b}-\mathbf{c}}{3}\]

    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 ,\[\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}\]                 \[=\left( \mathbf{a}-\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} \right)+\left( \mathbf{b}-\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} \right)+\left( \mathbf{c}-\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3} \right)\]                 \[=\frac{1}{3}[2\mathbf{a}-\mathbf{b}-\mathbf{c}+2\mathbf{b}-\mathbf{a}-\mathbf{c}+2\mathbf{c}-\mathbf{a}-\mathbf{b}]=\mathbf{0}\].


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