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Question:

Show that the four points A, B, C, and D with position vectors 4^i+5^j+^k, −^j−^k, 3^i+9^j+4^k, and 4(−^i+^j+^k) respectively are coplanar.

Solution:

If Position vector of →A=4^i+5^j+^k →B=−^j−^k →C=3^i+9^j+4^k →D=4(−^i+^j+^k) Point →A, →B, →C, →D all coplanar if [→AB →AC →AD]=0—(1) So, →AB=P.V. of →B−P.V. of →A=−4^i−6^j−2^k →AC=P.V. of →C−P.V. of →A=−^i+4^j+3^k →AD=P.V. of →D−P.V. of →A=−8^i−4^j+3^k So, for [→AB →AC →AD]=∣∣∣∣∣∣−4−6−2−143−8−43∣∣∣∣∣∣ Expand along R1 →−4[12+12]+6[−3+24]−2[4+32]=−4[24]+6[21]−2[36]=−96+126−72=0 So, we can say that point A, B, C, D are coplanar.