 | Term | Definition |
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projection of u onto v |
proj u = (<u>⋅<v>)/(||<v>||²)<v> |
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projection of work |
W =proj F |
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dot product form of work |
<F> ⋅ <PQ> |
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cross product |
<u> x <v> = (u₂v₃ - u₃v₂)<i> - (u₁v₃ - u₃v₁)<j>+ (u₁v₂ - u₂v₁)<k> |
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commutativity of the triple scalar product |
<u> ⋅(<v> x <w> = (<u> x <v>) ⋅<w> |
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area of a parallelogram |
||<u> x<v>|| were <u> and <v> are adjacent sides |
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volume of a parallelepiped |
V = |<u> ⋅(<v> x <w>)| where <u>, <v>, and <w> are adjacent sides |
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parametric equations of a line in space |
x = x₁ + at, y = y₁ + bt, z = z₁ + ct for point (x₁,y₁,z₁) and parallel vector <a,b,c> |
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symmetric equations of a line |
(x - x₁)/a = (y - y₁)/b = (z - z₁)/c for point (x₁,y₁,z₁) and parallel vector <a,b,c> |
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standard equation of a plane in space |
a(x - x₁) + b(y - y₁) + c(z - z₁) = 0 for point (x₁,y₁,z₁) and normal vector <a,b,c> |
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general equation of a place |
ax) + by + cz +d = 0 for normal vector <a,b,c> |
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angle between two planes |
cosθ = |<n₁>⋅<n₂>|/(||<n₁>|| * ||<n₂>||) |
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distance between a point and a plane |
D =|proj <PQ>| = |<PQ>⋅<n>|/||<n>|| where P is in the plane, Q is the point, and n is the plane normal |
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distance between a point and a plane |
D = |ax₀+by₀+cz₀+d|/√(a² +b² +c²) |
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distance between a point and a line in space |
D = ||<PQ> x <u> ||/||u|| where Q is a point in space, P is point on the line, and <u> is in the direction of the line |
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ellipsoid |
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 |
z²/c² - x²/a² - y²/a² = 1 |
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elliptic cone |
x²/a² + y²/b² - z²/c² = 0 |
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elliptic paraboloid |
z = x²/a + y²/b² |
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hyperbolic paraboloid |
z = y²/b² - x²/a² |
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surface of revolution revolved around the x-axis |
y² + z² = [r(x)]² |
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surface of revolution revolved around the y-axis |
x² +z² = [r(y)]² |
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surface of revolution revolved around the z-axis |
x² + y² = [r(z)]² |
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cylindrical to rectangular coordinates |
x = rcosθ, y = rsinθ, z = z |
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rectangular to cylindrical coordinates |
r² = x² + y², tanθ = y/x, z = z |
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spherical to rectangular coordinates |
x = ρsinφcosθ, y=ρsinφsinθ, z =ρcosφ |
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rectangular to spherical coordinates |
ρ² = x² + y² + z², tanθ = y/x, φ = arccos(z/(√(x² + y² + z²)) |
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spherical to cylindrical coordinates |
r² = ρ²sin²φ, θ=θ, z=ρcosφ |
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cylindrical to spherical coordinates |
ρ = √(r² + z²), θ = θ, φ = arccos (z/(√(r² +z ²)) |
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