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37 Cards in this Set

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DISTANCE FORMULA |p1, p2|
√(x2 - x1)² + (y2 - y1)² + (z2 - z1)²
2 minus 1
Equation of a Sphere with center (h,k,l) and radius r
(x - h)² + (y - k)² + (z - l)² = r²
vector
a quantity that has both magnitude and direction
vector components
<a1, a2, a3>
Definition of vector addition
If a and v are vectors positioned so the initial point of v is at the terminal point of a then the sum u + v is the vector from the initial point of u to the terminal point of v.
PQ + QR = PR
QS - PS = QP
Properties of vectors
a + b = b + a
a + (b + c) = (a + b) + c
a + 0 = a
a + -a = 0
scalar-c(a + b) = ca + cb
scalars(c + d)a = ca + da
scalars(cd)a = c(da)
1a = a
standard basis vectors
i <1,0,0>
j <0,1,0>
k <0,0,1>
unit vectors
a vector whose length is 1
(standard basis vectors)
resultant force
the vector sum of several forces acting on an object
position vector
the representation of a vector from the origin.
Given 2 points
a = <x2 - x1, y2 - y1, z2 - z1>
Magnitude or length of a vector
|a| = √a1² + a2² + a3²
Adding vectors algebraically
a + b = <a1 + b1, a2 + b2, a3 + b3>
dot product
a ∙ b = (a1)(b1) + (a2)(b2) + (a3)(b3)
Properties of the dot product
a ∙ a = |a|²
a ∙ b = b ∙ a
a ∙ (b + c) = a ∙ b + a ∙ c
(ca) ∙ b = c(a ∙ b) = a ∙ (cb)
0 ∙ a = 0
Theorem 3 If Θ is the angle between vectors a and b
a ∙ b = |a| |b| cos Θ
cos^-1 Θ = a ∙ b ∕ |a| |b|
Scalar projection of b onto a
comp(a) b = a ∙ b / |a|
vector projection of b onto a
proj a b = (a ∙ b / |a|²) ( a)
Work/Force/Distance Equation
W = F ∙ D if no angle
or
w = |F||D| cos Θ
Theorem 4 If Θ is the angle between a and b then
|a x b| = |a| |b| sin Θ

or sin^-1 Θ = |a x b| / |a| |b|
x²/a² + y²/b² + z²/c² = 1
Ellipsoid
(3)2nd order terms and a constant)
z²/c² = x²/a² + y²/b²
Cone
(3)2nd order variables
No constants.
If put in form x²/a² + y²/b² - z²/b² = 0 - Opens towards axis of negative variable.
z/c = x²/a² + y²/b²
Elliptic paraboloid
(2)2nd order terms that have same sign.
(1)1st order term.
No constants.
Opens along 1st order term axis.
2nd order terms determines + or -.
z/c = x²/a² - y²/b²
Hyperbolic Paraboloid
(2)2nd order terms that have opposite signs.
(1)1st order term.
No constants.
Neg 2nd order term = head of horse.
1st order term = top of saddle.
x²/a² + y²/b² - z²/c² = 1
Hyperboloid of one sheet
(3)2nd order terms
(1) constant
(1) negative term - opens along this term
-x²/a² - y²/b² + z²/c² = 1
Hyperboloid of two sheets
(3) 2nd order terms
(1) constant
(1) positive term - opens along this term
x² + y² = 1
Cylinder
(2) variables
extruded along missing variable
vector equation of a line
r = rօ + t(v)
r = (x,y,z) + t<a,b,c>
parametric equations
x = xօ + at
y = yօ + bt
z = zօ + ct
symmetric equations
(x - xօ)/a = (y - yօ)/b = (z - zօ)/c
vector equation of a plane
n ∙ r = n ∙ rօ
or
n ∙(r - rօ) = 0
scalar equation of a plane
a(x - xօ) + b(y - yօ) + c(z - zօ) = 0
normal vector
vector orthogonal to a plane
cross product
linear equation of a plane
ax + by + cz + d = 0
parallel planes
normal vectors are parallel
Distance from point to a plane
|ax₁ + by₁ + cz₁ + d|/√a² + b² + c²
Area of a parallelogram
|a x b|
volume of a parallelepiped
|a ∙ (b x c)|