JEE MAIN - Mathematics (2003 - No. 33)
Consider points $$A, B, C$$ and $$D$$ with position
vectors $$7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \widehat i - 3\widehat j + 4\widehat k$$ and $$5\widehat i - \widehat j + 5\widehat k$$ respectively. Then $$ABCD$$ is a :
vectors $$7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \widehat i - 3\widehat j + 4\widehat k$$ and $$5\widehat i - \widehat j + 5\widehat k$$ respectively. Then $$ABCD$$ is a :
parallelogram but not a rhombus
square
rhombus
None
Explanation
$$A = \left( {7, - 4,7} \right),B = \left( {1, - 6,10} \right),$$
$$C = \left( { - 1, - 3,4} \right)$$ and $$D = \left( {5, - 1,5} \right)$$
$$AB = \sqrt {{{\left( {7 - 1} \right)}^2} + {{\left( { - 4 + 6} \right)}^2} + {{\left( {7 - 10} \right)}^2}} $$
$$ = \sqrt {36 + 4 + 9} = 7$$
Similarly $$BC = 7,\,\,CD = \sqrt {41} ,\,\,DA = \sqrt {17} $$
$$\therefore$$ None of the options is satisfied
$$C = \left( { - 1, - 3,4} \right)$$ and $$D = \left( {5, - 1,5} \right)$$
$$AB = \sqrt {{{\left( {7 - 1} \right)}^2} + {{\left( { - 4 + 6} \right)}^2} + {{\left( {7 - 10} \right)}^2}} $$
$$ = \sqrt {36 + 4 + 9} = 7$$
Similarly $$BC = 7,\,\,CD = \sqrt {41} ,\,\,DA = \sqrt {17} $$
$$\therefore$$ None of the options is satisfied
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