CHAPTER 2 THE DOT CROSS PRODUCTS AND Then all lie in the required plane, so that    PP1 r r1, P2 P1 r2 r1, PP3 r3 r1 r  r2  r3  0 r1 r1 r1 x  ˆ   y z 2ˆ 1 1k 3 2k 4 0      i.e ˆ ˆ 3ˆ ˆ k 11x 5 13z 30 or  y   VECTOR TRIPLE PRODUCTS THREE VECTORS OF     If , and are three vectors, then and are called vector  A B C A C triple products. iˆ kˆ ˆj A  B  C  A1iˆ  A2  A3kˆ  B1 B2 B3 ˆj C1 C2 C3    B2C3  ˆ B1C3  B1C2  B3C2 B3C1 ˆ k iˆ kˆ ˆj  B2C3 B3C2 B3C1 B1C3 B1C2 B2C1 A1 A2 A3 A2 A1 B1C2  B3C1 iˆ B1C2  B2C3 ˆj  B2C1 A3 B1C3 B2C1 A3 B3C2 B3C1  B2C3 k  A1 B1C3 A2 B3C2 Also we have  A2 ˆ  B C B1C2 A3 B1C3 A2C1B2 A3C1B3 B2   A1C1  B2 A3C3 C2 A1B1 C2 A3 B3 ˆj B3 k   A1C1 B3 A2C2 C3 A1B1 C3 A2 B2 and so we have shown that   .  A B C A A Now since we interchange the roles of in the  B C    B A, B and C above argument and have A A  B C    B  BC A     BA  C AB  C and as such we use the above relations to easily show that
CHAPTER 2 THE DOT AND CROSS PRODUCTS Then PP1  r r1, P2 P1  r2 r1, PP3  r3 r1 all lie in the required plane, so that r r1  r2 r1  r3 r1  0 i.e x 2ˆ y  1 ˆ  z 1k  ˆ  3 ˆ 2k  3ˆ  4 ˆ  k  0 or 11x  5 y  13z  30 VECTOR TRIPLE PRODUCTS OF THREE VECTORS If A , B and C are three vectors, then A    and   C are called vector triple products. iˆ A  B  C  A1iˆ  A2 ˆj  A3kˆ  B1 ˆj B2 kˆ B3 C1 C2 C3    B2C3 B3C2 ˆ B1C3 B3C1  ˆ  B1C2 k  iˆ ˆj kˆ  A1 B2C3 B3C2 A2 B3C1 B1C3 A3 B1C2 B2C1  A2 B1C2 B2C1  A3 B3C1 B1C3 iˆ A1 B1C2 B2C1  A3 B2C3 B3C2 ˆj  A1 B3C1 B1C3  A2 B2C3 B3C2 k Also we have B C  A2 B1C2  A3 B1C3 A2C1B2 A3C1B3 ˆ  B2 A1C1  B2 A3C3 C2 A1B1 C2 A3 B3  ˆj  B3 A1C1  B3 A2C2 C3 A1B1 C3 A2 B2 k and so we have shown that A    B C . Now since A  B C   A  B we interchange the roles of A, B and C in the above argument and have A  B C   A  B   BC  A  BA  C AB  C  and as such we use the above relations to easily show that