Position vector of 4 points are given as −−→OA=2^i+3^j−^k,−−→OB=^i−2^j+3^k,−−→OC=3^i+4^j−2^k and −−→OD=^i−6^j+6^k
and O be the origin. Then,
A
all the 4 points are vertices of the tetrahedron with volume 8√3
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B
all the 4 points are coplaner
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C
−−→AB,−−→AC,−−→AD are orthogonal vectors
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D
all the 4 points are vertices of the tetrahedron with volume 4√3
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Solution
The correct option is B all the 4 points are coplaner Given −−→OA=2^i+3^j−^k −−→OB=^i−2^j+3^k −−→OC=3^i+4^j−2^k
and −−→OD=^i−6^j+6^k −−→AB=−^i−5^j+4^k−−→AC=^i+^j−^k−−→AD=−^i−9^j+7^k
Now [−−→AB−−→AC−−→AD]=∣∣
∣∣−1−5411−1−1−97∣∣
∣∣=2+30−32=0
So all the 4 points are coplaner