Exam 13: Vector Geometry

Let be the line . A) Find the distance between any point on and the origin as a function of B) Find the point on closest to the origin by minimizing .

Find a scalar such that the vectors and are orthogonal.

Let . Calculate the following.

Express as a linear combination of and .

For which values of h is the intersection of the horizontal plane and the ellipsoid empty?

Use cylindrical coordinates to describe the following set given in rectangular coordinates: .

Consider the 3 lines. Which of the following statements is correct?

Find the point of intersection of the three medians in the triangle with vertices , and .

Three nonzero vectors, in 3-space are in one plane (coplanar) if:

is a triangle and are the midpoints of , and respectively. A) Find the vectors , and in terms of and B) Compute the sum of the vectors in (A).

Write the equation of the quadric surface in standard form, and identify the type of the quadric conic.

Let A) Find the area of the parallelogram spanned by and . B) Find a vector orthogonal to and so that the volume of the parallelepiped spanned by , , and is .

Compute the area of the projection of the parallelogram with vertices , , , and on the xy-plane.

Express as a linear combination of the two vectors and .

Let be the line and be the plane . A) Find the point of intersection of and . B) Find an equation of the plane (in scalar form) that passes through the point and is orthogonal to .

Which of the following lines is parallel to the plane

Describe the trace obtained by intersecting the quadric surface with the plane .

The line is determined by the points and . The line is determined by the points and . Find a unit vector along the line perpendicular to both and .

Describe the trace obtained by intersecting the quadric surface with the plane

Find the angle between the vectors and .