CHAPTER 2 THE DOT CROSS PRODUCTS AND We let be the position vector of point P and Q, the terminal point of . Since r B  is perpendicular to 0 gives the equation of PQ  B r A, A  or r A  B A the plane. In rectangular coordinates, we have xiˆ 2iˆ iˆ 2iˆ    3  6kˆ   5   3  6kˆ yˆj zkˆ ˆj ˆj ˆj 2x 3y 6z 35 or    The distance from the origin to the plane is the projection of on . Thus, a unit B A A vector aˆ in the direction of is   . A iˆ kˆ ˆj 72 73 76  A    5 Then the projection of B on A  B a   . Example 2.10 P Q Given that 3iˆ 2kˆ and 2 4kˆ are position vectors of points and A    B  iˆ ˆj ˆj respectively, a) Find an equation for the plane passing through Q and perpendicular to PQ. line Z P R Q A r B O Y X Let position vector of some point R on the plane be . r 3 6kˆ   Then PQ OQ OP ˆj Since the plane is perpendicular to PQ Then
CHAPTER 2 THE DOT AND CROSS PRODUCTS We let r be the position vector of point P and Q, the terminal point of B . Since PQ  B r is perpendicular to A,  A  0 or r A  B A gives the equation of the plane. In rectangular coordinates, we have xiˆ  yˆj  zkˆ 2iˆ  3 ˆj  6kˆ  iˆ  5 ˆj  2iˆ  3 ˆj  6kˆ or 2x  3y  6z  35 The distance from the origin to the plane is the projection of B on A . Thus, a unit vector aˆ in the direction of A is A A  72 iˆ  73 ˆj  76 kˆ . Then the projection of B on A  B a      5 . Example 2.10 Given that A  3iˆ  ˆj  2kˆ and B  iˆ 2 ˆj 4kˆ are position vectors of points P and Q respectively, a) Find an equation for the plane passing through Q and perpendicular to line PQ. Z P R Q A r B O Y X Let position vector of some point R on the plane be r . Then PQ  OQ OP  3 ˆj 6kˆ Since the plane is perpendicular to PQ Then