) ( ) OB OA AC OC OA AD OD OA We have ] AB AC AD Therefore, the three vectors AB AC AD are coplanar and hence the four points , , A B C and D lie on a plane. Vector - - Applications of Vector Algebra Example . If the vectors , , a b c are coplanar, then prove that the vectors b b c c are also coplanar. Since the vectors , , a b c are coplanar, we have [ , ] .
a b c = Using the properties of the scalar triple product, we get ] b b c c = [ , ] [ , ] a b c c b b c c = [ , , ] [ , , ] [ , , ] [ , , ] a b c a c c b b c b c c = [ , , ] [ , , ] [ , , ] [ , , ] [ , , ] [ , , ] [ , , ] [ , , ] a b c a b a a c c a c a b b c b b a b c c b c a = [ , , ] [ , , ] [ , , ] a b c a b c a b c Hence the vectors b b c c are coplanar. Example . If , , a b c are three vectors, prove that [ ] [ , , ] c a b a a b c . Using theorem .
, we get ] c a b a = [ , , ] a b c = [ , , ] a b c .