CHAPTER 2 THE DOT CROSS PRODUCTS AND Example 2.5 Show that the vectors 3iˆ 2 , 3 5kˆ, 2iˆ 4kˆ form a right-       A ˆj kˆ B iˆ ˆj C ˆj angled triangle. This can be shown in two ways. Either we show that the sum of two vectors equals the third or the sum of the three vectors is identically zero. By trial and error we find that   , so that the vectors do form a triangle. A B C 0 On determining the dot products of the vectors in turn we discover that  . A C ⊥and the triangle is a right-angled triangle. Thus and C are A Example 2.6 Find the angles that the vector 3iˆ 6 2kˆ   makes with the coordinate axis. We A ˆj  ,  , x,y,z let the angles be that makes with the positive axes respectively. A 2 then    3 and A i ˆ 73  cos    64.60 Thus 73 Similarly    cos 1490   76 76 72  and  cos   73.40 72 Example 2.7 Find the projection of the vector  2  on the vector  4iˆ 4  7kˆ . A iˆ kˆ B ˆj ˆj A P  Q B S R Projection of cos where is the unit vector in the     A on B RS PQ A Q A bˆ bˆ direction of . B
CHAPTER 2 THE DOT AND CROSS PRODUCTS Example 2.5 Show that the vectors A  3iˆ 2 ˆj  kˆ , B  iˆ 3 ˆj  5kˆ, C  2iˆ  ˆj 4kˆ form a right- angled triangle. This can be shown in two ways. Either we show that the sum of two vectors equals the third or the sum of the three vectors is identically zero. By trial and error we find that A  B  C , so that the vectors do form a triangle. On determining the dot products of the vectors in turn we discover that A C  0 . Thus A and C are ⊥and the triangle is a right-angled triangle. Example 2.6 Find the angles that the vector A  3iˆ 6 ˆj  2kˆ makes with the coordinate axis. We let the angles be  ,  , that A makes with the positive x,y,z axes respectively. then and Thus 2 A ˆ   i  3 cos  73   73  64.60 Similarly and cos   76   76  1490 cos   72  72  73.40 Example 2.7 Find the projection of the vector A  iˆ 2 ˆj  kˆ on the vector B  4iˆ 4 ˆj  7kˆ . A P  Q R S B Projection of A on B  RS  PQ  A cos Q  A bˆ where bˆ is the unit vector in the direction of B .