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Midterm

# MATH 32A Study Guide - Midterm Guide: Parallelogram, Dot Product, Cross ProductExam

Department
Mathematics
Course Code
MATH 32A
Professor
All
Study Guide
Midterm

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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
 
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 spheres 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.