Page 31 - 35Linear Algebra

1.5 Review Problems 31

Remember that the cross product of two 3-vectors is given by

0
     0 0 
x x yz − zy
0
y × y := zx − xz .
   0   0 
0
z z 0 xy − yx 0
Indeed, 3-vectors are special, usually vectors an only be added, not
multiplied.
Lets find the force F (a vector) one must apply to a wrench lying along
   
1 0
1
the vector r =   ft, to produce a torque  
0 ft lb:
0 1
 
a
b .
(a) Find a solution by writing out this equation with F =  
c
(Hint: Guess and check that a solution with a = 0 exists).
 
1
(b) Add   to your solution and check that the result is a solution.
1
0
(c) Give a physics explanation of why there can be two solutions, and
argue that there are, in fact, infinitely many solutions.
(d) Set up a system of three linear equations with the three compo-
nents of F as the variables which describes this situation. What
happens if you try to solve these equations by substitution?

3. The function P(t) gives gas prices (in units of dollars per gallon) as a
function of t the year (in A.D. or C.E.), and g(t) is the gas consumption
rate measured in gallons per year by a driver as a function of their age.
The function g is certainly different for different people. Assuming a
lifetime is 100 years, what function gives the total amount spent on gas
during the lifetime of an individual born in an arbitrary year t? Is the
operator that maps g to this function linear?

4. The differential equation (DE)

d
f = 2f
dt

31