THE DOT CROSS PRODUCTS AND Hence CHAPTER 2 x y z 2iˆ 3 1iˆ 2 4kˆ 0      ˆj ˆj 2x 3y 6z 28 0     b) What is the distance from the point (-1,1,1) to the plane? (x1,y1,z1) The distance from the point to the plane is given by:   ax1 by1 cz1 d  distance   a b c 2 2 2 Where ax+by+cz=d is the equation of the plane. Thus the distance (-1,1,1) from to the plane is 5  2 1  3  6  28 Z 22 32 62   7  THE CROSS PRODUCT For vectors in space, we normally define axes of reference by the ‘right-hand rule’. OX, OY, OZ form a right-handed set if rotation from OX to OY takes a right-handed corkscrew OZ. action along the positive direction of Similarly, OY OZ rotation from to gives right-hand corkscrew X OX action along the positive direction of etc. y Definition: The cross or vector product of and written and read as ‘ A B A  B A sin  cross ’ is defined as a vector having the magnitude , where B AB  is the angle between the two vectors. The product vector acts in a direction perpendicular to and in such a sense that , , and A B A B A form a right-handed system-in that order.  B sin    A B AB  A B  B
CHAPTER 2 Hence THE DOT AND CROSS PRODUCTS 2iˆ 3 ˆj x 1iˆ  y  2 ˆj  z  4kˆ  0 2x  3y  6z  28  0 b) What is the distance from the point (-1,1,1) to the plane? The distance from the point (x1,y1,z1) to the plane is given by: distance  ax1  by1  cz1 d a 2  b 2  c 2 Where ax+by+cz=d is the equation of the plane. Thus the distance from (-1,1,1) to the plane is 2 1  3  6  28 22  32  62  7  5 Z THE CROSS PRODUCT For vectors in space, we normally define axes of reference by the ‘right-hand rule’. OX, OY, OZ form a right-handed set if rotation from OX to OY takes a right-handed corkscrew action along the positive direction of OZ. Similarly, rotation from OY to OZ gives right-hand corkscrew action along the positive direction of OX etc. X y Definition: The cross or vector product of A and B written A  B and read as ‘ A cross B ’ is defined as a vector having the magnitude AB sin  , where  is the angle between the two vectors. The product vector acts in a direction perpendicular to A and B in such a sense that A , B , and A  B form a right-handed system-in that order. A  B A  B  AB sin   B