CHAPTER 2 THE DOT CROSS PRODUCTS AND If happens to be in the same plane as , then will be perpendicular to A B and C A   . Hence . cos 90 0 B  C A A B  C    Therefore, three vectors are coplanar if . , 0 A B and C A Example 2.16 The vectors are coplanar since   2 3kˆ ,  2iˆ  2kˆ  3iˆ  A iˆ B and C kˆ ˆj ˆj ˆj 1 2 3 2 1 2 0 A B  C   3 1 1 Example 2.17 Find the volume of a parallelepiped whose edges are represented by 2iˆ 3 4kˆ , 2 3iˆ 2kˆ A   B  iˆ  kˆ and C   ˆj ˆj ˆj 2 3 4 Volume of parallelepiped is 1 2 1 7    A B C 3 1 2 Example 2.18 Find the constant a such that the vectors are 2iˆ , 2 3kˆ 3iˆ 5kˆ     ˆj kˆ iˆ ˆj and aˆj coplanar. Solution:      If coplanar then 0 3 i.e. 2 3  0 a 5 2 1 a=-4. from which we obtain Example 2.19 Find an equation for the plane determined by the points P1(2,-1,1), P2(3,2,-1) and P3(-1,3,2). P1, P2, P3 P(x,y,z) The position vectors of and any point are respectively
CHAPTER 2 THE DOT AND CROSS PRODUCTS If A happens to be in the same plane as B and C , then A will be perpendicular to B  C . Hence A   A B  C cos 90  0 . Therefore, three vectors A , B and C are coplanar if A   0 . Example 2.16 The vectors A  iˆ  2 ˆj 3kˆ , B  2iˆ ˆj  2kˆ and C  3iˆ  ˆj kˆ are coplanar since 1 2 A B  C  2 1 3 2  0 3 1 1 Example 2.17 Find the volume of a parallelepiped whose edges are represented by A  2iˆ 3 ˆj  4kˆ , B  iˆ  2 ˆj kˆ and C  3iˆ ˆj  2kˆ 2 3 Volume of parallelepiped is A B  C  1 2 3 1 4 1  7 2 Example 2.18 Find the constant a such that the vectors 2iˆ ˆj  kˆ , iˆ  2 ˆj 3kˆ and 3iˆ  aˆj  5kˆ are coplanar. Solution: If coplanar then      0 i.e. 2 1 3 a 5 2 3  0 from which we obtain a=-4. Example 2.19 Find an equation for the plane determined by the points P1(2,-1,1), P2(3,2,-1) and P3(-1,3,2). The position vectors of P1, P2, P3 and any point P(x,y,z) are respectively