Exam 12: Vectors and the Geometry of Space

A woman walks due west on the deck of a ship at mi/h. The ship is moving north at a speed of mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

An ellipsoid is created by rotating the ellipse about the x-axis. Find the equation of the ellipsoid.

Find the cross product .

A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is cm long. Find the magnitude of the torque about P correct to two decimal places. Let N, , .

Find the distance between the planes.

Find an equation of the plane that passes through the line of intersection of the planes and is perpendicular to the plane

Let and let u be a vector with length that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector .

Which of the given lines is parallel to the line

Find parametric equations for the line through and parallel to the vector

Find parametric equations for the line through and

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes , , and .

Suppose you start at the origin, move along the x-axis a distance of units in the positive direction, and then move downward a distance of units. What are the coordinates of your position?

Write inequalities to describe the solid upper hemisphere of the sphere of radius centered at the origin.

Find the center and radius of the sphere.

Find the point at which the line given by the parametric equations below intersects the plane.

Find an equation for the surface obtained by rotating the parabola about the y-axis.

Find an equation of the plane through the origin and parallel to the plane

Find an equation of the plane with x-intercept , y-intercept and z-intercept .

Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N W at a speed of km/h. (This means that the direction from which the wind blows is west of the northerly direction.) A pilot is steering a plane in the direction N E at an airspeed (speed in still air) of km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the ground speed of the plane. Round the result to the nearest hundredth.

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is times the distance from P to the yz-plane.