CHAPTER 2 THE DOT CROSS PRODUCTS AND In the case where two vectors are in the same direction, the angle between them is zero. a .b  cos 0  So a ab ab ~ ~ b On writing the two vectors and in their component form, i.e. A B A  A1iˆ  A2  A3kˆ ˆj B  B1iˆ  B2  B3kˆ ˆj    A B  Then    A1B1ˆ A1B2 A1B3 k ˆ ˆ ˆ ˆ    A2 B1 ˆj iˆ A2 B2 ˆj ˆj A2 B3 ˆj kˆ    A3 B1k ˆ A3 B2k ˆ A3 B3k k 0 1     iˆ iˆ kˆ kˆ Now ˆj ˆj 90 0 Also iˆ  iˆ kˆ  kˆ   ˆj ˆj We are left with    A B A1B1 A2 B2 A3 B3 Example 2.3 2iˆ 2 6iˆ 3 2kˆ     A ˆj kˆ and B ˆj If Then cos     A B AB A1B1 A2 B2 A3 B3 3.7 cos 12 6 2 4     4 cos Thus A B and the angle between A and B is  214  790 Example 2.4 Determine the values of a so that 2 2aiˆ 4kˆ are A  aiˆ  kˆ and B   aˆj ˆj ⊥if their dot product is zero. perpendicular (⊥). are A and B Thus 2a 4 0  aiˆ 2aiˆ 2   kˆ aˆj 2 ˆj 4kˆ 2a  or
CHAPTER 2 THE DOT AND CROSS PRODUCTS In the case where two vectors are in the same direction, the angle between them is zero. a So a .b  ab cos 0  ab ~ ~ b On writing the two vectors A and B in their component form, i.e. A  A1iˆ  A2 ˆj  A3kˆ B  B1iˆ  B2 ˆj  B3kˆ Then Now Also A B      A1B1ˆ ˆ  A1B2 ˆ ˆ  A1B3 ˆ k  A2 B1 ˆj iˆ  A2 B2 ˆj ˆj  A2 B3 ˆj kˆ  A3 B1k ˆ  A3 B2k ˆ  A3 B3k k iˆ iˆ  ˆj ˆj  kˆ kˆ  0  1 iˆ ˆj  iˆ kˆ  ˆj kˆ  90  0 We are left with A B  A1B1  A2 B2  A3 B3 Example 2.3 If Then A  2iˆ  2 ˆj kˆ and B  6iˆ 3 ˆj  2kˆ A B  AB cos  A1B1  A2 B2  A3 B3  3.7 cos  12 6 2  4 Thus A B  4 and the angle between A and B is cos  214  790 Example 2.4 Determine the values of a so that A  aiˆ 2 ˆj  kˆ and B  2aiˆ  aˆj 4kˆ are perpendicular (⊥). A and B are ⊥if their dot product is zero. Thus aiˆ 2 ˆj  kˆ 2aiˆ  aˆj 4kˆ  2a 2 2a 4  0 or