exam one review Flashcards
distance between two points
d = sq[(x-x1)^2+(y-y1)^2+(z-z1)^2]
standard equation of sphere
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
midpoint formula
C = (x1+ y1)/2 , (x2+ y2)/2 , (x3 +y3)/2
dot product
*two vectors: result is scalar
*the result is a sum (add up vectors)
*use dot product to find angle between two vectors
angle between two vectors
cos theta = u * v/
|u| |v|
orthogonal vectors
u dot v = 0
|vector projection| =
scalar projection
projection is used for:
*breaking down a force into its components and (find a force in a direction of motion)
scalar projection/ comp =
||u||
projection vector
|u dot v|
————- * u
||u||^2
work equation
W = mag of F * mag of PQ *cos theta
cross product
*two vectors results in a another vector
- n = u X v
- is anticommunative
purpose of cross product
- finding a vector orthogonal to 2 other vectors
- finding area of parallelogram & parallelopiped
- the cross product of two vectors are parallel = 0
area of some figure using cross product
- 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||
volume of a parallelpiped
V = | u dot (v X w) |
= triple scalar product
LINE SHIT
equation of a line in space
vector equation of a line
r = r0 +tv
*(x,y,z) = (x0,y0,z0) +tv
parametric equations
x = x0 +ta , y = y0 +tb ,
z = z0+tc
- vector v = ( a, b, c)
- point p = (x0, y0, z0)
symmetric equation of a line
x-x0 y-y0 z-z0
—— = ——– = ——-
a b c
- vector v = ( a, b, c)
- point p = (x0, y0, z0)
distance between a point and line
d = || PQ X V ||
—————–
|| v ||
PLANE SHIT
equation of plane
n dot PQ = 0
scalar equation of a plane
a(x-x0)+b(y-0)+c(z-z0)=0
*(a,b,c) = normal vector
*x0,y0,z0 = point
distance between a point and a plane
d = mag of proj. of QP onto n = | QP dot n |/ ||n||
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
find angle between two planes
cos theta =
|n1 dot n2|/
||n1|| ||n2||
distance from a point and plane
sq(a^2 +b^2+c^2)
*use a point in one plane and use the other plane for (a,b,c)
chap 3 shit
vector valued function
r(t) = f(t) i + g(t) j
*comp form: r(t) = (f(t), g(t))
finding domain of r(t)
*find domain of each function and find common domain
domain of cos and sin
all real numbers
domain of tan and sec
not defined at pi/2
plane curve
2-D
space curve
3-D
limits
lim t->a (r(t) = L
derivatives of vector valued functions
d/dt r(t) = lim tri.t-> 0
r(t+(tri.t))-r(t) /tri.t
unit tangent vector
T(t) = r’(t)/ ||r’(t)||
ARC LENGTH FORMULAS
plane curve (2d)
s = a->b sq[((f’(t)^2) + (g’(t)^2)] dt
space curve (3d)
s = a->b sq[((f’(t)^2) + (g’(t)^2)+(h’(t)^2)] dt
arc length function
s(t) = a->t sq[((f’(u)^2) + (g’(u)^2)] du
CURVATURE
k = curvature
curvature formula
k = ||T’(s)||
alternative curvature formula
k = || T’(t)||/ ||r’(t)||
alternative curvature (f(x) = y)
k = y’’/ [1+(y’)^2]^3/2
NORMAL VECTOR
N(t) = T’(t)/ ||T’(t)||
*basically the unit vector of T’(t)
BINORMAL VECTOR
B(t) = T(t) X N(t)
