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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=OBOA

=^j^k(4^i+5^j+^k)

=4^i6^j2^k

AC=OCOA

=3^i+9^j+4^k(4^i+5^j+^k)

=^i+4^j+3^k

AD=ODOA

=^4i+4^j+4^k(4^i+5^j+^k)

=8^i^j+3^k

Now,
[AB,AC,AD] =∣ ∣462143813∣ ∣

=4(12+3)+6(3+24)+(2)(1+32)

=60+12666=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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