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 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

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!)

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