Problem 1. (a) .
Problem 2. (a)
(c ) .
Problem 3. skipped.
Problem 4. (a) The plane is given by .
(b) The plane is given by .
(c ) This sphere has as its radius and has as its center . Its equation is then
(d) To find the equation of this sphere we need to find its center and we need to find its radius. Its center point is the mid-point or average of the two given points and so
The radius can be found by computing the distance between the two given points and dividing it by 2:
We can now use our sphere equation to find an equation for this sphere.
Problem 5. For the function we cannot plug in any pairs that make the expression negative. This means we need
This inequality defines a region in the plane that is a filled in circle of radius , centered at the origin. If we call this region , then we have that the domain of is .
Problem 6. (a)
(b) Since one unit vector is
(c ) . The components of are
(d) In general, given a unit vector parallel to is
Note that this works provided (otherwise we’d be trying to divide by ).
Problem 7. (skipped)