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Calc 3 Exam 1
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Gravity
Terms in this set (25)
Length of a vector v = < a, b, c >
√(a²+b²+c²)
Unit vector
e_v = (1 / ‖v‖ ) v
Triangle Inequality Theorem
‖v + w‖ ≤ ‖v‖ + ‖w‖
Distance Formula in R^3
The distance | P - Q | between the points
P = (a1, b1, c1) and Q = (a2, b2, c2) is...
|P-Q| = √(a₂-a₁)²+(b₂-b₁)²+ (c₂-c₁)²
Equation of a sphere centered at Q = (a, b, c) is...
(x-a)² + (y-b)²+(z-c)² = R²
Equation of the right circular cylinder in R^3 of radius R whose central axis is the vertical line through (a, b, 0) is...
(x-a)² + (y-b)² = R²
v • v =
‖v‖²
Law of Cosines
v • w = ‖v‖ ‖w‖ cos(θ)
v and w are perpendicular iff
v • w = 0
The projection of u along v is the vector
u || v
(u • v) / (v • v) * v
The vector u that is orthogonal to v is...
u⊥v = u - u‖v
Vector decomposition
u = u‖v + u⊥v
v x w is _____ to v and w
orthogonal
v x w has length
‖v‖ ‖w‖ sin(θ)
w x v = ___ x ___
- v x w
Area of parallelogram spanned by v and w =
‖v x w‖
Equation of a plane
Plane through P = (x, y, z) with normal vector n = <a, b, c>
n • < x, y, z >
Ellipsoid
(x/a)²+(y/b)²+(z/c)²=1
Hyperboloids
One Sheet
(x/a)²+(y/b)²= (z/c)²+1
Two Sheet
(x/a)²+(y/b)²= (z/c)²-1
Elliptic Cone
(x/a)²+(y/b)²= (z/c)²
Paraboloids
Elliptic
z = (x/a)² + (y/b)²
Hyperbolic
z = (x/a)² - (y/b)²
Cylindrical to Rectangular
x =
y =
z =
x = r cosθ
y = r sinθ
z = z
Rectangular to Cylindrical
r =
theta =
z =
r = √ x² + y²
tanθ = y / x
z = z
Spherical to Rectangular
x =
y =
z =
x = p sinφ cosθ
y = p sinφ sinθ
z = pcosφ
Rectangular to Spherical
p =
theta =
phi =
p = √x²+y²+z²
tan θ = y/x
cosφ = z/p
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