**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!)