Department

MathematicsCourse Code

MATH 32AProfessor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**1 pages of the document.**2014 Midterm 1 (Clean)

Problem 1:

Let be a nonzero two-dimensional vector.

A. (10 pts) Find a vector, unique up to a scale factor of , which is orthogonal to and has the same length

as .

B. (10 pts) If

is a nonzero vector and is a scalar, find the vector which points in the same

direction as

and has length.

Problem 2:

In it is sometimes useful to consider the vectors

A. (15 pts) Compute and .

B. (5 pts) Let

and

denote two 3D vectors where , , , and are scalars and

, are the vectors described above. Compute the dot product

and express your answer only in terms

of the quantities you see in this problem.

Problem 3:

(20 pts) Recall that a parallelogram with equal sides is called a rhombus (as you may have learned in high school).

Using vector algebra, show that the two segments connecting the two pairs of opposite vertices of a rhombus (aka

its diagonals) intersect at a right angle. (Hint: Draw a picture and label some points.)

Problem 4:

Suppose and

, where is an angle less than

.

A. (5 pts) Noting that the angle between these two vectors is , express as a “polynomial” in terms

of and .

B. (5 pts) Now regard and

as 3-D vectors with their -component equal to zero. Use the cross product to

express as a “polynomial” in terms of and .

Problem 5:

(15 pts) In let and

be two vectors and consider the points in that satisfy

Show that these points form a sphere and identify that sphere’s radius and center.

Problem 6:

(15 pts) If and are distinct points in space, show that the points equidistant from both and form a plane,

and identify its normal vector.

Problem 7:

(15 pts) Consider two planes, both of which pass through the origin:

In , not all of , , are equal. Find an equation for the line formed by the intersection of both of these planes.

Verify explicitly that the line you have lies on both planes.

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