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FIRST HOUR MAKEUP EXAM MATH 223, Section 3

Instructor: Kris Green, February 9, 1998

1.

The plane z = 3x - 2y + 4 lies in 3-space.

(a)

Find the equation of the plane parallel to this plane passing through the point (1, -1, 0).

(b)

What value(s) of t will make the plane -2x + ty - (t² -3)z = 5 perpendicular to the first two planes?

2.

For the function f(x,y) = e^-x e^1-y

(a)

Sketch cross sections of z = f(x,y) with x fixed.

(b)

Sketch cross sections of z = f(x,y) with y fixed.

(c)

Sketch a contour diagram of z = f(x,y) in the region $-3 \le x \le 3, -3 \le y \le 3$.

(d)

If a drop of water is placed on the surface at x = y = 1 and always rolls down the steepest path, sketch the path of the water droplet with a dotted line on your contour diagram.

3.

For the vector field

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

(a)

Find the equation of the surface in 3-space where the vector field has a constant magnitude of 1.

(b)

Describe the shape of the curves made where this surface touches the xy plane, the xz plane, and the yz plane.

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, decompose the vector $\vec{F}(P)$ as the sum of two vectors, one parallel to and one perpendicular to $\vec{v}$.

5.

A geometrical solid is made from a top and bottom (both hexagonal) and six side faces which are all squares. The diagram shows the object, along with a normal vector to each face. Opposite faces are parallel, and each face meets both the top and bottom at right angles.

(a)

Express the vectors $\vec{v}_4, \vec{v}_5, \vec{v}_6$ in terms of the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$.

(b)

Express $\vec{v}_7$ as a function of the vectors $\vec{v}_1, \dots, \vec{v}_6$. You may not need all six of the other vectors to do this.

(c)

If $\vec{v}_7 = 3\hat{i} - 2 \hat{j} + \hat{k}$, what is $\vec{v}_8$ as a unit vector?