Math 32 A Practice Midterm 1 1 Determine whether the points P Q R lie on a straight line a P 2 4 2 Q 3 7 2 R 1 3 3 b P 0 5 5 Q 1 2 4 R 3 4 2 2 Suppose a and b are non zero vectors a Under what circumstances is compa b compb a a Under what circumstances is proja b projb a 3 Find the angle between the diagonal of a cube and one of its faces 4 Find an equation for the plane defined by points equidistant from the points 1 0 2 and 3 4 0 5 Consider the points P such that the distance from P to the point A 1 5 3 is twice the distance to B 6 2 2 Show that the set of all such points is a sphere 6 Consider a projectile shot with initial velocity v0 above a slope with angle from the horizontal How long does it take the projectile to hit the slope Write your answer in terms of gravitational constant g and v0 v0x v0y Figure 1 Problem 6 illustration 7 Consider an amusement park ride consisting of a large rotating wheel with rotating carriages that the passengers ride in If the large wheel rotates with angular velocity 1 radians per second i e as time varies from t 0 to t 1 an angle of 1 radians is swept out and the carriage rotates with angular velocity 2 write an equation for the position r t of a passenger s head assuming that the passenger boards the ride as shown in figure 2 Hint this is very similar to the cycloid trajectory see figure 2 below Here the position of the passenger s head is described similarly to the point on the rotating wheel for the cycloid 1 Figure 2 Problem 7 illustration The passenger s head is initially at r 0 r1 0 0 r2 2