Exam 11: Vectors and the Geometry of Space

A certain lunar crater is shaped approximately as a paraboloid (see figure). The cross sections of the crater parallel to the lunar surface are circular, with a maximum radius of 60 meters. The depth of the crater is 9 meters. Find an equation for the surface of the crater. [Write your equation such that the bottom of the crater is located at (x, y, z) = (0, 0, 0). Assume the z-axis is perpendicular to the lunar surface.]

Find two unit vectors orthogonal to the two given vectors.

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

Compute the given determinant.

Find the distance between the given objects. The planes x - 8y + z = 20 and x - 8y + z = -57

Find comp_b a for the given vectors.

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

Find two unit vectors orthogonal to the two given vectors.

Determine whether the vectors a and b are parallel.

Determine whether the vectors a and b are parallel.

Find the indicated volume. Volume of the parallelpiped with three adjacent edges formed by , , and

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 13,

Find comp_b a for the given vectors.

If find a vector such that

Use the Cauchy-Schwartz Inequality in n dimensions to show that , where m is an odd natural number.

Find an equation of the given plane. The plane containing the point and perpendicular to the planes 5x + 5y - 2z = -8 and -5x - y - 5z = -8

Compute ||2a - 4b||.

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

Find the indicated area. Area of the triangle with vertices , , and

Find symmetric equations of the line through and .