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FIRST HOUR EXAM, MATH 223, Section 3

Instructor: Kris Green, February 11, 1997

1.

Multiple Choice (5 points each).

(a)

The component of the vector $\vec{v} = 5\hat{i} - 2\hat{j} - 4\hat{k}$ in the direction of the vector $\vec{w} = 2\hat{i} + \hat{j} - \hat{k}$ is

$$2\vec{w}$$ b. 2

$$-2\vec{w}$$ d. -2

(b)

The contours of the function $z = f(x,y) = \frac{y}{x^2}$ are shaped like:

a. concentric circles b. parabolas

c. hyperbolas d. parallel lines

(c)

If f(x,y) is a linear function then the statement below is

f(4,5) - f(4,7) = f(2,2) - f(2,4)

a. always true b. always false

c. true under certain conditions d. not enough information given

(d)

Two planes z = ax+by+c and z=mx+ny+d are parallel if (more than one answer may be correct):

a. their normal vectors are either 0 or 180 degrees apart

b. their normal vectors are perpendicular

c. the constants c and d are equal

d. the quantities ax + by and mx + ny are equal for all x and y

2.

Given the vectors $\vec{v} = -\hat{i} + 2\hat{j} + 5\hat{k}$ and $\vec{w} = 3\hat{i} + \hat{j} - \hat{k}$ find:

(a)

$\vec{v} + 3\vec{w}$

(b)

$\vec{v} \cdot \vec{w}$

(c)

$\vec{v} \times \vec{w}$

(d)

The angle, $\theta$, between $\vec{v}$ and $\vec{w}$

3.

(a)

The following table represents values for a linear function of the variables x and y. Find the equation for this linear function.

-1 1

X 2 14 12

3 18

(b)

Find a unit normal vector to the plane whose equation you found in above.

4.

Let

$$f(x,y)=\frac{1}{(x-2)^2 + (y-1)^2}$$

(a)

Draw the contours f(x,y) = c for c=0,1,2,3.

(b)

What is the domain (for both x and y) of the function?

(c)

Describe the contours f(x,y) = c as $c \rightarrow \infty$.

(d)

Use the contour diagram to decide if f is an increasing, decreasing or constant function of the variable x at the point x=0, y=0.5.

5.

Suppose a Star Fury and a Shadow vessel are in combat near the space station Babylon 5. Put the space station at the origin of a coordinate system. The Star Fury is at position (-5,10,10) and has a velocity vector of $\vec{v}_g = 10\hat{i} - 50\hat{j} -10\hat{k}$. The Shadow vessel is at position (30,-40,0) and has velocity $\vec{v}_b = -3\hat{i} + 4\hat{j} + 0\hat{k}$.

(a)

If the Star Fury fires an energy weapon (speed = $\infty$) at the Shadow vessel, what is the direction in which the pilot should fire? This should be a unit vector.)

(b)

Write the velocity of the Star Fury as a sum of two vectors, one along the direction of the Shadow vessel's velocity and one perpendicular to the Shadow vessel's velocity.

(c)

Find the equation of the plane containing the vectors $\vec{v}_g$ and $\vec{v}_b$.