Multivariable Calculus: Chapter 11

Magnitude of a Vector :

Slope of a Vector : Given the vector A=<a1, a2>,Slope=a2/a1

Cross Product of Vectors (given an angle) :

Unit Vectors : i <1,0,0>j <0,1,0>k <0,0,1>

How to find the unit vectors tangent and normal to a curve : 1) Find the derivative of the equation of the curve2) Plug in a given point to the derivative to find the slope at the given instant3) Determine a vector using the slope found (a2/a1)4) Find the magnitude of the vector and divide the vector by the magnitude to get the direction5) For Normal: find the negative reciprocal of the slope and determine direction using new vector

Direction of a Vector :

How to find the angle between two vectors using dot product :

Properties of Dot Products :

Equation of a Plane : ax+by+cz=d

Vector Equation for a Line : Let P(x,y,z) be an arbitrary point on line LLet r0 & r be the position vectors of P0 & Pr=<x0+ta, y0+tb, z0+tc>r=<x,y,x>, r0=<x0, y0, z0>, v=<a, b, c>

Finding parametric equations of a line given two points it passes through : 1) Find the vector between the two points given2) The vector you find is the vector parallel to the line3) Use one of the points given to construct the parametric equations with the vector

How to show that three vectors are coplanar : triple scalar product is equal to 0

Parametric Equation for a Line L :

Symmetric Equations : Solve each parametric equation for a line L for t, as shown

Circle : The set of all points a set distance (r) from a given point (center)

Parabola (vertex equation and definition) : The set of all points equidistant from a point (focus) and a line (directrix)

Hyperbola (labeled picture) :

Hyperbola Asymptotes (Horizontal/x-oriented) :

Cartesian Coordinates to Cylindrical Coordinates :

Spherical Coordinates : (ρ, Φ, θ)Where rho (ρ) is the radius of the sphere (distance out from the origin, can ONLY be POSITIVE), phi (Φ) is the angle with the positive z-axis (0≤Φ≤π), and theta (θ) is the angle from the positive x-axis

Cylinder : a surface composed of all the lines parallel to a given line that passes through a given plane curve (called the generating curve of the cylinder)

How to draw a hyperboloid of one sheet : The variable that is NEGATIVE is the axis along which the shape runsHyperboloid of one sheet is a continuous shape, no space between the two halves of the shape and an ellipse is present at the symmetry line1) Set the negative variable equal to 0 and draw the ellipses formed2) Set one of the other two variables equal to 0 and draw the hyperbolas on the plane of the two variables that are not being set to 03) Set the other variable equal to 0 and draw the hyperbolas on the plane of the two variables not set to 0

Hyperboloid of two sheets :

Sphere :

How to draw a circular paraboloid : The variable with no square is the axis along which it runs (the variable that the circle equation is set equal to)1) Set the variable that is not squared equal to 0 and draw a circle in the plane of the two variable it includes2) Set one of the other two variables equal to 0 and draw the parabola3) Set the last variable equal to 0 and draw the parabola

Elliptical Cone :

Converting between rectangular/cylindrical/spherical :

Converting from spherical to rectangular :