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Vectors

indicate magnitude and direction

vector between (x,y) and (a,b)

<a-x, b-y>

unit vector

magnitude of 1, indicates direction

magnitude of a vector

(x²+y²)^0.5

to find the unit vector of a vector <x,y>

<x/((x²+y²)^0.5), y/((x²+y²)^0.5)>

dot product of <a, b> and <x,y>

ax+by

u dot v is equal to

|u||v| costheta

cosine of the angle between vectors u and v

(u dot v) / (|u||v|)

if dot product is zero, vectors are

orthogonal

component subscript v of u definition

the scalar part of u in the direction of v

component subscript v of u formula

(u dot v) / |v|

projection subscript v of u definition

the vector part of u in the direction of v

projection subscript v of u formula

((u dot v) / (|v|²)) v

if (ax+b)/(cx+d), the end behavior as x→ infinity

y approaches a/c from top if y>c, from bottom if y<c

local behavior for zeroes

when x is around that, y is (other factors with x subbed in for them)(original one)

vertical asymptotes are determined by

denominator

slant asymptotes happen when

the highest exponent in the numerator of a rational function is exactly one greater than the highest exponent in the denominator

point discontinuities occur when

(x-a) is in both the numerator and denominator at x=a

u cross v equals

|u||v|sintheta

(u cross v)sin theta is zero when

u and v are parallel

what is (ucrossv)sin theta also equal to

magnitude of parallelogram formed

paralleliped

slanty box, scalar triple product, zero if vectors are coplanar

STP

a dot (b x c)

area of tetrahedron

1/6 * STP

linear transformation T satisfies

T(c(vectoru))=c(T(vectoru)) for all vectors, T(u+v)=Tu+Tv

linear transformations

rotation, reflection, dilation, any combination of the former

non-linear transformation

translation

the column length of S must

equal the row length of T in ST

conditions for gauss jordan elimination

pivot of a row is the leftmost nonzero entry, any zero row is below every nonzero row, all pivots are to the right of above pivots, any column with a pivot is above and below the pivot, the pivots are one

determinant of a 3 by 3 matrix

a(ei-fh)-b(di-gf)+c(dh-eg)

determinant

cross out row and column, subtract the left to right by the right to left

if a column has no pivot

it’s a free variable

reflection matrix over y=ax

[cos2theta sin2theta
sin2theta -cos2theta]

rotation matrix (CCW)

[costheta -sintheta

sintheta costheta]

focus

c units away from the center in the direction of the semimajor/semitransverse axis

focus of an ellipse

a²-b²=c²

eccentricity of an ellipse

c/a=PF(distance from point to focus)/PD(distance from point to directrix)

when c of an ellipse is 0

it’s a perfect circle

focus of a hyperbola

c²=a²+b²

when e of a hyperbola is closer to 1, it’s

skinnier

asymptotes of a hyperbola

b/a if horizontal, a/b if vertical

vertex of a parabola

(h,k)

directrix of a parabola

y=k-p OR x=h-p

focus of a parabola

y=k+p OR x=h+p

eccentricity of a parabola

1

parabola equation

4p(y-k)=(x-h)²

when p is positive for a vertical parabola

it opens upwards

when p is positive for a horizontal parabola

it opens rightwards

degenerate hyperbola equation

(hyperbola equation)=0

degenerate hyperbola

lines are the asymptotes

degenerate parabola

(ax+b)²=0 or (ax+b)(ax+c)=0 or no real points satisfy eq

when is a degenerate parabola a single line?

(ax+b)²=0

when is a degenerate parabola 2 parallel lines

(ax+b)(ax+c)=0

when is there no real graph for a degenerate parabola

no real points satisfy the equation

degenerate circle

point at the center, = 0

parameter equation for ellipses

x(t)=h+acost
y(t)=k+bsint

helpful trig relationships for parametric equations

sin²theta+cos²theta=1, 1+tan²theta=sec²theta, 1+cot²theta=cocsec²theta

coshx

(e^x+e^-x)/2

sinhx

(e^x-e^-x)/2

cosh²x+sinh²x=

cosh2x

cosh²x-sinh²x

1

domain of coshx and sinhx

all real numbers

range of coshx

[1, infinity)

range of sinhx

(-infinity, infinity)

coshx+sinhx

e^x

non-continuous growth eq

P(1+r/n)^nt

continuous growth eq

Pe^rt

in partial fraction decomposition, if a factor in the denominator is to a power, the coverup method

only yields the value for the higher power

term n of an arithmetic series

𝑎1+(𝑛−1)d

sum of an arithmetic series

n/2(a1+an)

term n of a geometric series

a1*r^(n-1)

sum of a geometric series

a1(1-r^n)/(1-r)

sum of an infinite geometric series

a1/(1-r)

summation of k

(n²+n)/2

summation of k²

n(n+1)(2n+1)/6

summation of k³

n²(n+1)²/4