Calculus II, Chapt. 13
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- DISTANCE FORMULA |p1, p2|
- √(x2 - x1)² + (y2 - y1)² + (z2 - z1)²
- 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
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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 - 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 - standard basis vectors
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i <1,0,0>
j <0,1,0>
k <0,0,1> - unit vectors
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a vector whose length is 1
(standard basis vectors) - resultant force
- the vector sum of several forces acting on an object
- position vector
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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
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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 - 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| - 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
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W = F ∙ D if no angle
or
w = |F||D| cos Θ - 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| - x²/a² + y²/b² + z²/c² = 1
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Ellipsoid - z²/c² = x²/a² + y²/b²
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Cone - z/c = x²/a² + y²/b²
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Elliptic paraboloid - z/c = x²/a² - y²/b²
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Hyperbolic Paraboloid - x²/a² + y²/b² - z²/c² = 1
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Hyperboloid of one sheet - -x²/a² - y²/b² + z²/c² = 1
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Hyperboloid of two sheets - x² + y² = 1
- Cylinder
- vector equation of a line
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r = rÖ… + t(v)
r = (x,y,z) + t<a,b,c> - parametric equations
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x = xÖ… + at
y = yÖ… + bt
z = zÖ… + ct - symmetric equations
- (x - xÖ…)/a = (y - yÖ…)/b = (z - zÖ…)/c
- vector equation of a plane
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n ∙ r = n ∙ rօ
or
n ∙(r - rօ) = 0 - scalar equation of a plane
- a(x - xÖ…) + b(y - yÖ…) + c(z - zÖ…) = 0
- normal vector
- vector orthogonal to a plane
- linear equation of a plane
- ax + by + cz + d = 0
- parallel planes
- normal vectors are parallel
- Distance from point to a plane
- |ax₠+ by₠+ cz₠+ d|/√a² + b² + c²
- Area of a parallelogram
- |a x b|
- volume of a parallelepiped
- |a ∙ (b x c)|