exam one review Flashcards

1
Q

distance between two points

A

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

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2
Q

standard equation of sphere

A

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

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3
Q

midpoint formula

A

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

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4
Q

dot product

A

*two vectors: result is scalar

*the result is a sum (add up vectors)

*use dot product to find angle between two vectors

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5
Q

angle between two vectors

A

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

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6
Q

orthogonal vectors

A

u dot v = 0

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7
Q

|vector projection| =

A

scalar projection

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8
Q

projection is used for:

A

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

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9
Q

scalar projection/ comp =

A

||u||

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10
Q

projection vector

A

|u dot v|
————- * u
||u||^2

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11
Q

work equation

A

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

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12
Q

cross product

A

*two vectors results in a another vector

  • n = u X v
  • is anticommunative
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13
Q

purpose of cross product

A
  • finding a vector orthogonal to 2 other vectors
  • finding area of parallelogram & parallelopiped
  • the cross product of two vectors are parallel = 0
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14
Q

area of some figure using cross product

A
  • 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||
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15
Q

volume of a parallelpiped

A

V = | u dot (v X w) |

= triple scalar product

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16
Q

LINE SHIT

A
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17
Q

equation of a line in space

A
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18
Q

vector equation of a line

A

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

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19
Q

parametric equations

A

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

  • vector v = ( a, b, c)
  • point p = (x0, y0, z0)
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20
Q

symmetric equation of a line

A

x-x0 y-y0 z-z0
—— = ——– = ——-
a b c

  • vector v = ( a, b, c)
  • point p = (x0, y0, z0)
21
Q

distance between a point and line

A

d = || PQ X V ||
—————–
|| v ||

22
Q

PLANE SHIT

23
Q

equation of plane

A

n dot PQ = 0

24
Q

scalar equation of a plane

A

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

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

25
distance between a point and a plane
d = mag of proj. of QP onto n = | QP dot n |/ ||n||
26
two plane line intersection
*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
27
find angle between two planes
cos theta = |n1 dot n2|/ ||n1|| ||n2||
28
distance from a point and plane
ax0 +by0+cz0+k ------------------- sq(a^2 +b^2+c^2) *use a point in one plane and use the other plane for (a,b,c)
29
chap 3 shit
30
vector valued function
r(t) = f(t) i + g(t) j *comp form: r(t) = (f(t), g(t))
31
finding domain of r(t)
*find domain of each function and find common domain
32
domain of cos and sin
all real numbers
33
domain of tan and sec
not defined at pi/2
34
plane curve
2-D
35
space curve
3-D
36
limits
lim t->a (r(t) = L
37
derivatives of vector valued functions
d/dt r(t) = lim tri.t-> 0 r(t+(tri.t))-r(t) /tri.t
38
unit tangent vector
T(t) = r'(t)/ ||r'(t)||
39
ARC LENGTH FORMULAS
40
plane curve (2d)
s = a->b sq[((f'(t)^2) + (g'(t)^2)] dt
41
space curve (3d)
s = a->b sq[((f'(t)^2) + (g'(t)^2)+(h'(t)^2)] dt
42
arc length function
s(t) = a->t sq[((f'(u)^2) + (g'(u)^2)] du
43
CURVATURE
k = curvature
44
curvature formula
k = ||T'(s)||
45
alternative curvature formula
k = || T'(t)||/ ||r'(t)||
46
alternative curvature (f(x) = y)
k = y''/ [1+(y')^2]^3/2
47
NORMAL VECTOR
N(t) = T'(t)/ ||T'(t)|| *basically the unit vector of T'(t)
48
BINORMAL VECTOR
B(t) = T(t) X N(t)