6-30. If A=A 1 ˆe 1 +A 2 ˆe 2 +A 3 eˆ 3 and B=B 1 ˆe 1 +B 2 eˆ 2 +B 3 ˆe 3 show that A×B =−B×A.
6-31. If A×B = 0 and B×C= 0 , then calculate A×C . Justify your answer.
6-32. (a) Find the equation of the plane which passes through the points
P 1 (3 , 10 ,13) P 2 (0, 11 ,12) P 3 (5 , 12 ,14).
(b) Find the perpendicular distance from the origin to this plane.
(c) Find the perpendicular distance from the point (6, 3 ,18) to this plane.
6-33. Show that the rules for calculating the moment of a force about a line L
can be altered as follows: If r is the position vector from a point Pon the line Lto
any point on the line of action of the force F , then M =r ×F is the moment about
point P on the line Land M ·ˆeLis the moment about the line L, where eˆLis a unit
vector in the direction of L.
6-34. A force F= 100( ˆe 1 + 2 ˆe 2 − 2 ˆe 3 )lbs acts at the point P 1 (2 , 2 ,4).
(a) Find the moment of F about the origin.
(b) Find the moment of F about the point P 2 (− 1 , 3 ,−4).
(c) Find the moment about the line passing through the origin and the point P 2.
6-35. Find the indefinite integral of the following vector functions
(a) u (t) = tˆe 1 +ˆe 2 −t^2 ˆe 3 (b) u (t) = tˆe 1 + sin tˆe 2 + cos tˆe 3
6-36. Find the position vector and velocity of a particle which has an acceleration
given by a = cos teˆ 1 + sin tˆe 2 if at time t= 0 the position and velocity are given by
r (0) = 0 and v (0) = 2 ˆe 3.
6-37. The acceleration of a particle is given by a =ˆe 1 +tˆe 2 .If at time t= 0 the
velocity is v =v (0) = ˆe 1 +ˆe 3 and its position vector is r =r (0) = ˆe 2 , then find the
velocity and position as a function of time.
6-38. Distance between parallel planes If (r −r 0 )·N = 0 and (r −r 1 )·N = 0 are the
equations of parallel planes, then show the distance between the planes is given by
the projection of r 1 −r 0 onto the normal vector N.
