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37 Cards in this Set
- Front
- Back
- 3rd side (hint)
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DISTANCE FORMULA |p1, p2|
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√(x2 - x1)² + (y2 - y1)² + (z2 - z1)²
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2 minus 1
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Equation of a Sphere with center (h,k,l) and radius r
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(x - h)² + (y - k)² + (z - l)² = r²
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vector
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a quantity that has both magnitude and direction
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vector components
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<a1, a2, a3>
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Definition of vector addition
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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 |
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Properties of vectors
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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 |
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standard basis vectors
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i <1,0,0>
j <0,1,0> k <0,0,1> |
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unit vectors
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a vector whose length is 1
(standard basis vectors) |
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resultant force
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the vector sum of several forces acting on an object
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position vector
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the representation of a vector from the origin.
Given 2 points a = <x2 - x1, y2 - y1, z2 - z1> |
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Magnitude or length of a vector
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|a| = √a1² + a2² + a3²
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Adding vectors algebraically
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a + b = <a1 + b1, a2 + b2, a3 + b3>
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dot product
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a ∙ b = (a1)(b1) + (a2)(b2) + (a3)(b3)
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Properties of the dot product
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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 |
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Theorem 3 If Θ is the angle between vectors a and b
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a ∙ b = |a| |b| cos Θ
cos^-1 Θ = a ∙ b ∕ |a| |b| |
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Scalar projection of b onto a
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comp(a) b = a ∙ b / |a|
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vector projection of b onto a
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proj a b = (a ∙ b / |a|²) ( a)
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Work/Force/Distance Equation
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W = F ∙ D if no angle
or w = |F||D| cos Θ |
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Theorem 4 If Θ is the angle between a and b then
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|a x b| = |a| |b| sin Θ
or sin^-1 Θ = |a x b| / |a| |b| |
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x²/a² + y²/b² + z²/c² = 1
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Ellipsoid
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(3)2nd order terms and a constant)
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z²/c² = x²/a² + y²/b²
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Cone
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(3)2nd order variables
No constants. If put in form x²/a² + y²/b² - z²/b² = 0 - Opens towards axis of negative variable. |
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z/c = x²/a² + y²/b²
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Elliptic paraboloid
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(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 -. |
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z/c = x²/a² - y²/b²
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Hyperbolic Paraboloid
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(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. |
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x²/a² + y²/b² - z²/c² = 1
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Hyperboloid of one sheet
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(3)2nd order terms
(1) constant (1) negative term - opens along this term |
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-x²/a² - y²/b² + z²/c² = 1
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Hyperboloid of two sheets
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(3) 2nd order terms
(1) constant (1) positive term - opens along this term |
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x² + y² = 1
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Cylinder
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(2) variables
extruded along missing variable |
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vector equation of a line
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r = rօ + t(v)
r = (x,y,z) + t<a,b,c> |
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parametric equations
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x = xօ + at
y = yօ + bt z = zօ + ct |
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symmetric equations
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(x - xօ)/a = (y - yօ)/b = (z - zօ)/c
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vector equation of a plane
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n ∙ r = n ∙ rօ
or n ∙(r - rօ) = 0 |
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scalar equation of a plane
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a(x - xօ) + b(y - yօ) + c(z - zօ) = 0
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normal vector
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vector orthogonal to a plane
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cross product
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linear equation of a plane
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ax + by + cz + d = 0
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parallel planes
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normal vectors are parallel
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Distance from point to a plane
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|ax₁ + by₁ + cz₁ + d|/√a² + b² + c²
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Area of a parallelogram
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|a x b|
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volume of a parallelepiped
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|a ∙ (b x c)|
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