exam one review

Question 1

distance between two points

Answer 1

d = sq[(x-x1)^2+(y-y1)^2+(z-z1)^2]

Question 2

standard equation of sphere

Answer 2

r^2 = (x-a)^2 + (y-b)^2 + (z-c)^2

*use distance formula to find radius if there is no common point

*(a,b,c) = center points

Question 3

midpoint formula

Answer 3

C = (x1+ y1)/2 , (x2+ y2)/2 , (x3 +y3)/2

Question 4

dot product

Answer 4

*two vectors: result is scalar

*the result is a sum (add up vectors)

*use dot product to find angle between two vectors

Question 5

angle between two vectors

Answer 5

cos theta = u * v/
|u| |v|

Question 6

orthogonal vectors

Answer 6

u dot v = 0

Question 7

|vector projection| =

Answer 7

scalar projection

Question 8

projection is used for:

Answer 8

*breaking down a force into its components and (find a force in a direction of motion)

Question 9

scalar projection/ comp =

Answer 9

||u||

Question 10

projection vector

Answer 10

|u dot v|
————- * u
||u||^2

Question 11

work equation

Answer 11

W = mag of F * mag of PQ *cos theta

Question 12

cross product

Answer 12

*two vectors results in a another vector

• n = u X v
• is anticommunative

Question 13

purpose of cross product

Answer 13

• finding a vector orthogonal to 2 other vectors
• finding area of parallelogram & parallelopiped
• the cross product of two vectors are parallel = 0

Question 14

area of some figure using cross product

Answer 14

• if two vectors are not given, then use a common point to find two lengths/vectors and then use cross product

(ie: A = || PQ X PR ||

• A = ||u X v||

Question 15

volume of a parallelpiped

Answer 15

V = | u dot (v X w) |

= triple scalar product

Question 16

LINE SHIT

Question 17

equation of a line in space

Question 18

vector equation of a line

Answer 18

r = r0 +tv
*(x,y,z) = (x0,y0,z0) +tv

Question 19

parametric equations

Answer 19

x = x0 +ta , y = y0 +tb ,
z = z0+tc

• vector v = ( a, b, c)
• point p = (x0, y0, z0)

Question 20

symmetric equation of a line

Answer 20

x-x0 y-y0 z-z0
—— = ——– = ——-
a b c

• vector v = ( a, b, c)
• point p = (x0, y0, z0)

Question 21

distance between a point and line

Answer 21

d = || PQ X V ||
—————–
|| v ||

Question 22

PLANE SHIT

Question 23

equation of plane

Answer 23

n dot PQ = 0

Question 24

scalar equation of a plane

Answer 24

a(x-x0)+b(y-0)+c(z-z0)=0

*(a,b,c) = normal vector
*x0,y0,z0 = point

Question 25

distance between a point and a plane

Answer 25

d = mag of proj. of QP onto n = | QP dot n |/ ||n||

Question 26

two plane line intersection

Answer 26

*equal both plane equations and solve for a variable

*plug variable into any equation and get another variable

• set z = t and find equations, then set t = to any number and find equations

Question 27

find angle between two planes

Answer 27

cos theta =
|n1 dot n2|/
||n1|| ||n2||

Question 28

distance from a point and plane

Answer 28

sq(a^2 +b^2+c^2)

*use a point in one plane and use the other plane for (a,b,c)

Question 29

chap 3 shit

Question 30

vector valued function

Answer 30

r(t) = f(t) i + g(t) j

*comp form: r(t) = (f(t), g(t))

Question 31

finding domain of r(t)

Answer 31

*find domain of each function and find common domain

Question 32

domain of cos and sin

Answer 32

all real numbers

Question 33

domain of tan and sec

Answer 33

not defined at pi/2

Question 34

plane curve

Answer 34

2-D

Question 35

space curve

Answer 35

3-D

Question 36

limits

Answer 36

lim t->a (r(t) = L

Question 37

derivatives of vector valued functions

Answer 37

d/dt r(t) = lim tri.t-> 0
r(t+(tri.t))-r(t) /tri.t

Question 38

unit tangent vector

Answer 38

T(t) = r’(t)/ ||r’(t)||

Question 39

ARC LENGTH FORMULAS

Question 40

plane curve (2d)

Answer 40

s = a->b sq[((f’(t)^2) + (g’(t)^2)] dt

Question 41

space curve (3d)

Answer 41

s = a->b sq[((f’(t)^2) + (g’(t)^2)+(h’(t)^2)] dt

Question 42

arc length function

Answer 42

s(t) = a->t sq[((f’(u)^2) + (g’(u)^2)] du

Question 43

CURVATURE

Answer 43

k = curvature

Question 44

curvature formula

Answer 44

k = ||T’(s)||

Question 45

alternative curvature formula

Answer 45

k = || T’(t)||/ ||r’(t)||

Question 46

alternative curvature (f(x) = y)

Answer 46

k = y’’/ [1+(y’)^2]^3/2

Question 47

NORMAL VECTOR

Answer 47

N(t) = T’(t)/ ||T’(t)||

*basically the unit vector of T’(t)

Question 48

BINORMAL VECTOR

Answer 48

B(t) = T(t) X N(t)