1) The three vectors are coplanar if their scalar triple product is zero.
2) The three vectors are coplanar if they are linearly dependent.
(10.1)
u⋅(v×w)=∣∣3−4481−111−1∣∣
=3∣∣1−11−1∣∣−8∣∣−441−1∣∣+1∣∣−441−1∣∣
=3(0)−8(0)+0=0 The following vectors lie in U.
Since the vector u,v and −v are linearly dependent, they lie in U.
(10.2)
u⋅(v×w)=∣∣3−47817110∣∣
=3∣∣1710∣∣−8∣∣−4710∣∣+1∣∣−4717∣∣
=3(−7)−8(−7)+(−35)=0 The following vectors lie in U.
Since the vector u,v and u−v are linearly dependent, they lie in U.
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