# Vector Homework 3

Problem 1.  Consider the function $f(x,y) = 2x + 3y$.

(a) What is the domain of this function?

(b) Draw a picture of the level set

$\displaystyle \left \{ \, (x,y) \, : f(x,y) = 0 \right \}.$

Problem 2.  Consider the vector $\vec{v} = \langle \, 2, 3 \, \rangle$.

(a) Draw a picture of the set

$\displaystyle \left\{\, \vec{w} \in \mathbb{R}^2 : \vec{w}\cdot\vec{v} = 0 \right\}.$

(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 $\mathbb{R}^3$: it detects when two vectors are parallel!  (Just like the dot product detects when two vectors are perpendicular).

In particular, if $\vec{v}$ and $\vec{w}$ are parallel vectors, then $\vec{v} \times \vec{w} = \vec{0}$.

The point of this problem is to have you explain why this is true.  To do this, first note that two vectors $\vec{v}$ and $\vec{w}$ are parallel if one is a scalar multiple of the other, that is if

$\vec{w} = c\vec{v}$

for some scalar $c \in \mathbb{R}$.  Next, write out the components of $\vec{v}$ and write out the components of $\vec{w}$ and apply our formula for $\vec{v} \times \vec{w}$ 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 10.  Write down the (scalar) equation of the plane that is parallel to the plane

$2x + 3y + 5z + 7 = 0$

but that passes through the point $(1, 1, 1)$.

Problem 11.  Find the equation for the plane that touches the sphere

$x^2 + y^2 + z^2 = 1$

at the point $(0, 0, 1)$ and only at this point.  (Hint: Draw a picture of the sphere first!)