MATH 32A Study Guide - Midterm Guide: Parallelogram, Dot Product, Cross ProductExam
Course CodeMATH 32A
This preview shows half of the first page. to view the full 1 pages of the document.
2014 Midterm 1 (Clean)
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
B. (10 pts) If
is a nonzero vector and is a scalar, find the vector which points in the same
and has length.
In it is sometimes useful to consider the vectors
A. (15 pts) Compute and .
B. (5 pts) Let
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.
(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.)
, 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 .
(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.
(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.
(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.
You're Reading a Preview
Unlock to view full version