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Question
Show that the four points A(4,5,1), B(0,−1,−1), C(3,9,4) and D(−4,4,4) are co-planar.
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Solution
We know that if four points A,B,C,D are coplanar when their co-terminous vector −−→AB,−−→AC and −−→AD will be coplanar,
∴[−−→AB,−−→AC,−−→AD]=0
Given A(4,5,1), B(0,−1,−1), C(3,9,4) and D(−4,4,4)
Considering O(0,0,0) as the initial point.
−−→OA=4^i+5^j+^k,−−→OB=−^j−^k,
−−→OC=3^i+9^j+4^k and −−→OD=−4^i+4^j+4^k
∴−−→AB=−−→OB−−−→OA
=−^j−^k−(4^i+5^j+^k)
=−4^i−6^j−2^k
−−→AC=−−→OC−−−→OA
=3^i+9^j+4^k−(4^i+5^j+^k)
=−^i+4^j+3^k
−−→AD=−−→OD−−−→OA
=−^4i+4^j+4^k−(4^i+5^j+^k)
=−8^i−^j+3^k
Now,
[−−→AB,−−→AC,−−→AD] =∣∣
∣∣−4−6−2−143−8−13∣∣
∣∣
=−4(12+3)+6(−3+24)+(−2)(1+32)
=−60+126−66=0
Therefore −−→AB,−−→AC and −−→AD are co-planar. These three vectors are co-intial vectors.
Hence, the points A(4,5,1), B(0,−1,−1), C(3,9,4) and D(−4,4,4) are co-planar.
∴[−−→AB,−−→AC,−−→AD]=0
Given A(4,5,1), B(0,−1,−1), C(3,9,4) and D(−4,4,4)
Considering O(0,0,0) as the initial point.
−−→OA=4^i+5^j+^k,−−→OB=−^j−^k,
−−→OC=3^i+9^j+4^k and −−→OD=−4^i+4^j+4^k
∴−−→AB=−−→OB−−−→OA
=−^j−^k−(4^i+5^j+^k)
=−4^i−6^j−2^k
−−→AC=−−→OC−−−→OA
=3^i+9^j+4^k−(4^i+5^j+^k)
=−^i+4^j+3^k
−−→AD=−−→OD−−−→OA
=−^4i+4^j+4^k−(4^i+5^j+^k)
=−8^i−^j+3^k
Now,
[−−→AB,−−→AC,−−→AD] =∣∣ ∣∣−4−6−2−143−8−13∣∣ ∣∣
=−4(12+3)+6(−3+24)+(−2)(1+32)
=−60+126−66=0
Therefore −−→AB,−−→AC and −−→AD are co-planar. These three vectors are co-intial vectors.
Hence, the points A(4,5,1), B(0,−1,−1), C(3,9,4) and D(−4,4,4) are co-planar.
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