A) vertices of an equilateral triangle
B) vertices of a right angled triangle
C) vertices of an isosceles triangle
D) collinear points
Correct Answer: D
Solution :
Given that, Position vector of \[A\,(\vec{a}\,-2\vec{b}+3\vec{c})\] Position vector of \[B\,(-2\vec{a}\,+3\vec{b}-\vec{c})\] Position vector of \[C\,(4\vec{a}\,-7\vec{b}+7\vec{c})\] Area of \[\Delta \,ABC\] \[=\frac{1}{2}\left| \begin{matrix} 1 & -2 & 3 \\ -2 & 3 & -1 \\ 4 & -7 & 7 \\ \end{matrix} \right|\] \[=\frac{1}{2}[1\,(21-7)+2(-14+4)+3\,(14-12)]\] \[=\frac{1}{2}\,[14-20+6]=0\] Hence, given points are collinear.
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