Exam 11: Vectors and the Geometry of Space

Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. Find the distance between the two insects at time . Round your answer to two decimal places.

An object is pulled 5 feet horizontally across a floor, using a force of 65 pounds. The direction of the force is above the horizontal. Find the work done. Round your answer to two decimal places.

Find the angle between the vectors for u and v given below.

A chute at the top of a grain elevator of a combine funnels the grain into a bin (see figure). Find the angle between two adjacent sides. Round your answer to one decimal place. ​

Identify the following quadric surface.

Given , find .

Convert the point from spherical coordinates to rectangular coordinates.

Find an equation in spherical coordinates for the equation given in rectangular coordinates. ​ ​

Given , find .

A child applies the brakes on a bicycle by applying a downward force of 20 pounds on the pedal when the crank makes a angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places. ​ ​

Suppose a toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a angle with the horizontal as shown in the figure below. Find the work done in pulling the wagon 50 feet. Round your answer to one decimal place. ​ ​

Find a unit vector u in the direction opposite of .

To carry a 500-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a angle away from the vertical and the other makes a angle as shown in the figure below. Find the vertical component of each worker's force. Round your answer to two decimal places. ​ ​ ​

Identify the equation of the following graph.

Find the coordinates of the midpoint of the line segment joining the points given below. (-6, 6, 1), (-4, 10, 11)

Find the set of parametric equations of the line through the point that is parallel to the line and . ​

Given and , find .

Find the length of the minor axis of the ellipse generated when the surface is intersected by the plane .

Suppose a 48,000-pound truck is parked on a slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force perpendicular to the hill. Round your answer to one decimal place.

Determine whether any of the lines given below are parallel or identical. ​ ​