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JCECE Engineering JCECE Engineering Solved Paper-2003

  • question_answer
    The area of triangle whose vertices are\[A(1,\,\,-1,\,\,2),\,\,\,B(2,\,\,1,\,\,-1)\]and\[C(3,\,\,-1,\,\,2)\]is:

    A) \[13\,\,sq\,\,unit\]                         

    B) \[\sqrt{13}\,\,sq\,\,unit\]

    C) \[56\,\,sq\,\,unit\]                         

    D)  \[\sqrt{6}\,\,sq\,\,unit\]

    Correct Answer: B

    Solution :

    If \[O\] be the point of origin, the position vectors of a vertices are                 \[\overset{\to }{\mathop{\mathbf{OA}}}\,=\widehat{\mathbf{i}}-\widehat{\mathbf{j}}+2\widehat{\mathbf{k}},\,\,\overset{\to }{\mathop{\mathbf{OB}}}\,=2\widehat{\mathbf{i}}+\widehat{\mathbf{j}}-\widehat{\mathbf{k}}\]                 \[\overset{\to }{\mathop{\mathbf{OC}}}\,=3\widehat{\mathbf{i}}-\widehat{\mathbf{j}}+2\widehat{\mathbf{k}}\] Now,     \[\overset{\to }{\mathop{\mathbf{AB}}}\,=2\widehat{\mathbf{i}}+\widehat{\mathbf{j}}-\widehat{\mathbf{k}}-(-\widehat{\mathbf{i}}-\widehat{\mathbf{j}}+2\widehat{\mathbf{k}})\]                        \[=\widehat{\mathbf{i}}+2\widehat{\mathbf{j}}-3\widehat{\mathbf{k}}\] and        \[\overset{\to }{\mathop{\mathbf{AC}}}\,=3\widehat{\mathbf{i}}-\widehat{\mathbf{j}}+2\widehat{\mathbf{k}}-(\widehat{\mathbf{i}}-\widehat{\mathbf{j}}+2\widehat{\mathbf{k}})\]                       \[=2\widehat{\mathbf{i}}\] \[\therefore \]Area of triangle\[=\frac{1}{2}||\overset{\to }{\mathop{\mathbf{AB}}}\,\times \overset{\to }{\mathop{\mathbf{AC}}}\,||\]                 \[=\frac{1}{2}\left| \left| \begin{matrix}    \widehat{\mathbf{i}} & \widehat{\mathbf{j}} & \widehat{\mathbf{k}}  \\    1 & 2 & -3  \\    2 & 0 & 0  \\ \end{matrix} \right| \right|\]                 \[=\frac{1}{2}|[\widehat{\mathbf{i}}(0)-\widehat{\mathbf{j}}(6)+\widehat{\mathbf{k}}(-4)]|\]                 \[=\frac{1}{2}|[-6\widehat{\mathbf{i}}-4\widehat{\mathbf{k}}]|\]                 \[=\frac{1}{2}\sqrt{36+16}=\frac{1}{2}\times 2\sqrt{13}\]                 \[=\sqrt{13}sq\,\,unit\] Note: If area of triangle is zero, then points are collinear.


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