Problem 1. Consider the function .
(a) What is the domain of this function?
(b) Draw a picture of the level set
Problem 2. Consider the vector .
(a) Draw a picture of the set
(b) What does this problem have in common with Problem 1 above?
Problem 3. Read pages 43-46 in our textbook and then complete Activity 9.18.
Problem 4*. Here is an interesting property concerning the cross product of two vectors in : it detects when two vectors are parallel! (Just like the dot product detects when two vectors are perpendicular).
In particular, if and are parallel vectors, then .
The point of this problem is to have you explain why this is true. To do this, first note that two vectors and are parallel if one is a scalar multiple of the other, that is if
for some scalar . Next, write out the components of and write out the components of and apply our formula for to find that their cross product is the zero vector.
Problem 5. Complete Exercise 2 from Section 9.4.
Problem 6. Complete Exercise 3a and 3b from Section 9.4.
Problem 7*. Complete Exercise 2 from Section 9.5
Problem 8. Complete Exercise 3a – 3e from Section 9.5
Problem 9. Write a poem about reading Section 9.5.
Problem 10. Write down the (scalar) equation of the plane that is parallel to the plane
but that passes through the point .
Problem 11. Find the equation for the plane that touches the sphere
at the point and only at this point. (Hint: Draw a picture of the sphere first!)