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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2000

  • question_answer
    If A, B, C are the vertices of a triangle whose position vectors are \[\vec{a},\vec{b},\vec{c}\]a and G the centroid of the \[\Delta A B C,\] then\[G\vec{A}+G\vec{B}+G\vec{C}=\]

    A)  \[\vec{0}\]                                        

    B)  \[\vec{A}+\vec{B}+\vec{C}\]

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

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

    Correct Answer: A

    Solution :

    Position vectors of vertices A, B and C of the triangle \[ABC=\vec{a},\vec{b},\vec{c}\] Position vector of centroid of the triangle \[G=\frac{\vec{a}+\vec{b}+\vec{c}}{3}\]therefore,                 \[G\vec{A}+G\vec{B}+G\vec{C}=\left[ \vec{a}-\frac{(\vec{a}+\vec{b}+\vec{c})}{3} \right]\]                                 \[+\left[ \vec{b}-\frac{(\vec{a}+\vec{b}+\vec{c})}{3} \right]\]                                 \[+\left[ \vec{c}-\frac{(\vec{a}+\vec{b}+\vec{c})}{3} \right]\] \[=\frac{1}{3}[2\vec{a}-\vec{b}-\vec{c}+2\vec{c}-\vec{a}-\vec{b}]\] \[=0\]


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