1. [6 points per part] Here are some short, unrelated plane problems.

(a) Find the plane through the points (1, 2, 3), (2, 3, 4), and (4, 6, 8).

(b) Find the line of intersection of the planes y = 2x+ 3 and 4x� 5y + 2z = 7.

Write your answer in parametric form.

ATEC 1,1 ,DA- B-

<AC→=(34,5 >

AI×Ac→=( I,-2,1 ) ~ normal vector

of plane

So : ( x-D - 2(y . 2) + (2--3)=0

or:

x-2ytz@LeIsarbitrarilysetx.t

.

So y=2t+3 ,

and 22=7-4×+52

÷ITI¥.IE?nIf

2. [2 points per part] Suppose u and v are 3D vectors that look like this:

That is, u points along the positive x-axis, and v is in the yz-plane.

(You don’t have to show work on this problem.)

x

y

z

u

v

Identify the following values as positive, negative, or zero. (Circle your answer.)

(a) u · v: positive negative zero

(b) The x-component of u⇥ v: positive negative zero

(c) The y-component of u⇥ v: positive negative zero

(d) The z-component of u⇥ v: positive negative zero

3. [7 points] Graph the polar curve r =1

sin(✓)� cos(✓). Please label the scales on your axes.

(Hint: This is a graph you’ve seen before. Don’t just draw a picture; do some algebra first.)

x

y

Y

0

o

oo

r( sine - cost )=ly

rsinO - nose -- /

y- ×=1

y=×+l- hey,,;tnh¥s y

- '

L

4. [8 points] You just got a job at Quadricorp, the company that builds quadric surfaces.

Your first assignment: make a hyperboloid of two sheets following these specifications:

• The hyperboloid “points” along the z-axis.

• The hyperboloid goes through the points (0, 0, 2), (3, 0, 4), and (6, 4, 8).

Give the equation of this hyperboloid.

5. [7 points] Consider the vector function r(t) = ht2 � 3t,pt+ 1, t2 + ti.

Let ` be the line tangent to the space curve of r(t) at the point (0, 2, 12).

Write parametric equations for `.

s¥s¥t÷¥i:c€÷E*⇒¥ - ¥+914

9¥@÷t¥ ¥ ¥

area,

Flt )=(2t - 3,

2¥,

2++1 ) so t =3

F' (3) ={ 3

,÷

,

7) ← direction of line

s.ie?zII#

6. [10 points] The acceleration of a friendly bee at time t is given by the vector function

a = h6, 6t, eti.

At time t = 0, the bee is at the origin. At time t = 2, the bee is at the point (4, 5, 6).

Where is the bee at time t = 3?

JG )={6++4,3+2+4,

ett G)F. ( t ) = ( 3+2+4++4

,

t3 + C at + C,

et + Gt + G)

FG ) - ( GCs ,

1+63=(0,0-0),

so §±gco =

-

1

FC2) ={ 12+24,8+2<2 ,

e 2+2<3 - D= { 4,5, 6)

soC

,

=- 4

Ca=¥

g= ¥

so: FG)=(35-43+3 - It

,

et + ( ¥ )t - 1)

F (3) =( 15,225 ,

e 3+(7243-1)

sothe bee is at C15.22.fi#+9@

7. I’ve got some vectors a and b. Here’s a picture:

a

b

✓

(a) [2 points] In the picture to the right, draw proja

�projba

�.

(Clearly label your work so I can see what’s going on.)

(b) [6 points] Suppose I know the following:

• a = h4, 2, 4i.• proja

�projba

�= h3, 32 , 3i.

• ✓ is acute.

What’s ✓?

:

projatf"¥) Y ;

it

projgaIaY-lF4l6-61prga.CprojbaTf9tFt9-4.5m@oycosO-4yI.andcosO-YsoasosyI.t

cos 'D = §

:- = §

F=c@f3)-Fdr@