b z c and, r x a y b z c , then , , p q r , , a b c Note By theorem . , if , , a b c are non-coplanar and ≠ then the three vectors p , x a y b z c q , x a y b z c and, r x a y b z c are also non-coplanar. Example . If , k b k c , find .
Solution: By the defination of scalar triple product of three vectors, = Vector - - Example . Find the volume of the parallelepiped whose coterminus edges are given by the vectors , k i We know that the volume of the parallelepiped whose coterminus edges are , , a b c is given by |[ , , ]| a b c . Here, , k b k c = + Since [ , , ] a b c , the volume of the parallelepiped is | | cubic units. Example .
Show that the vectors , i are coplanar. Here, , , k b k c = + We know that , , a b c are coplanar if and only if [ , , ] a b c = . Now, [ , , ] a b c Therefore, the three given vectors are coplanar. Example .
If , k i mj are coplanar, find the value of m . Since the given three vectors are coplanar, we have m m ⇒ = − . Example . Show that the four points ( , , ), ( , , ), ( , , ), ( , , ) lie on a same plane.
Let ( , , ), ( , , ), ( , , ), ( , , ) B C D . To show that the four points , , A B C D lie on a plane, we have to prove that the three vectors AB AC AD are coplanar. Now, AB (