Calculus 3: 3D Shapes

Sphere

x²/a² + y²/b² + z²/c² = r²

Cylinder

x²/a² + y²/b² = r²

Cone

z = +/- sqrt(x^2 + y^2

Ellipsoid

x²/a² + y²/b² + z/c = r² (a, b, c are different)

Parabloid

z/c = x²/a² + y²/b²

Elliptic Parabloid

z/c = x²/a² + y²/b² (a and b are different)

1 Sheet Hyperboloid

x²/a² + y²/b² - z²/c² = r²

2 Sheet Hyperboloid

x²/a² - y²/b² - z²/c² = r²

Parabolic Hyperboloid

z/c = x²/a² - y²/b²

Sphere

Cylinder

Cone

Ellipsoid

Elliptic Parabloid

1 sheet hyperboloid

2 sheet hyperboloid

parabolic hyperboloid

Dot Product of (a) * (b)

(a₁*b₁ + a₂b₂ + ...)

Work Formula

F * d

What does it mean if a*b = 0?

Vectors a and b are perpendicular

What does it mean if a x b = 0?

Vectors a and b are parallel

|a x b| = |a||b|sinθ

Length of Cross Product

Dot Product of a*b in terms of θ

|a||b|cosθ = (a) dot (b)

Area of a Parallelogram

|a x b|

Area of a Triangle

½| a x b|

Volume of a Parallelipiped

|a * (b x c)|

Scalar Triple Product

|a * (b x c)|

Volume of a Pyramid

1/6|a * (b x c)|

Equation of a Line

L = P₀ + (v)t

Parametric Form of a Line

x = x₀ + at

y = y₀ + bt

z = z₀ + ct

Symmetric Form of a Line

(x - x₀)/a = (y - y₀)/b = (z-z₀)/c

Vector Equation of a Plane

n * P₀P = 0

(n = normal vector,

P₀P = line in plane)

Scalar Equation of a Plane

a(x - x₀) + b(y - y₀) + c(z - z₀) = 0

or

ax + by + cz = d

Angle between two planes

cosθ = n*m / |n||m|

Equation of Plane Between 3 Points

Create a line from one point to a second point, then a second line from point to point three. Cross the two lines to generate a vector normal to the plane. Choose one of the three points and plug them into x₀,y₀,z₀.

Line of Intersection of Two Planes

Make the parametric equations of the planes equal to each other, solving for either t₁ or t₂. Substitute that value in to a second parametric equation for t₂.

Find the shortest distance between a point and a second point within a plane

Find a point on the plane P₀, make a line P-P₀. Project PP₀ onto the normal vector of the plane, giving the perpendicular distance from the point to the plane

Find the shortest distance between a point P and a second point on a line L.

Find a point P₀ on the line L with direction vector S. The equation for the shortest distance between the point P and the Line L is given by |PP₀ x S| / |s|.