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FIRST HOUR EXAM, MATH 223, Section 3

Instructor: Kris Green, February 6, 1998

1.

The line y = 2x - 5 and z = 0 lies in the xy-plane in 3-space.

(a)

Find the equation of the plane that contains this line and the point (1,-1,1).

(b)

At what point(s) does this plane touch the coordinate axes?

2.

Given the contour diagram below, answer the following questions.

(a)

Does the surface corresponding to this diagram seem to be represented by a function? Why or why not?

(b)

Imagine pouring water onto the surface at point Q. Sketch the path of the water's flow, assuming that the water flows downhill along the steepest path.

(c)

Sketch a graph of the two sections x = 2 and y = 1.

3.

For the vector field

$$\vec{G}(x,y,z) = \sqrt{z} \hat{i} + 3x \hat{j} - 2y \hat{k},$$

describe the set of all points (x,y,z) where $\vert\vert\vec{G}(x,y,z)\vert\vert = 1$.You may want to sketch contour or section plots to help you.

4.

The curve shown below has a tangent vector

$$\vec{v} = -\hat{i} + 2\hat{j} - \hat{k}$$

at the point P = (1, -1, 3). The vector field $\vec{F}(\vec{r}) = xy \hat{i} + yz \hat{j} - xz \hat{k}$ occupies the same space as the curve. At point P, what is the angle between the vector field and the curve?

5.

A cube is oriented randomly in 3-space with one face, labeled S₁ given by a portion of the plane x - 2y + 3z = 5. The adjacent face, S₂ has a normal vector $\vec{v}_2 = -6\hat{i} + 3\hat{j} + 4\hat{k}$.

(a)

What is the vector $\vec{v}_1$ expressed as a unit vector?

(b)

Give the vectors $\vec{v}_3$ and $\vec{v}_4$. Use the fact that the faces of a cube are at right angles to each other.

(c)

Using the fact that the faces of a cube meet at right angles, find the vectors $\vec{v}_5$ and $\vec{v}_6$.