Unit 1

Unit 1

1. Navigating 3-Space

   • going from (x,y) to (x,y,z)
   • Right Hand Rule
     • Put your index finger along the x axis, middle finger along the y axis, and your thumb will always point in the z direction
   • Important Objects in 3 Dimensions
     • xy plane: z=0 xz plane: y=0 yz plane: x=0

2. Linear Functions

   • Z=mx+ny+c
     • m= Dz/Dx n=Dz/Dy
   • Finding the Equation of a Linear Function
     • If given three points, solve for m and n -Make sure when figuring out what points to use, make sure m has the same x coord. while n has the same y coord.
   • Recognizing Linear Functions
     • Cross Sections
       • all equally spaced, parallel lines
     • Contour Diagrams
       • Parallel lines
     • Tables
       • check and make sure all changes in the x,y,z directions are constant or are approx. constant

3. Cross Sections

   • In Y
     • -set x equal to a constant -create a table to find the equations of the cross sections -graph the functions 
   • In X
     • -set y equal to a constant -create a table to find the -equations of the cross sections graph the functions 

4. Contour Diagrams

   • -set z equal to a constant, c -create a table where you set c equal to a number and solve for y to get an equation -graph the equation

5. Vectors

   • Finding the Length of a Vector
     • for the vector v=ai+bj+ck ||v||= sqrt(a^2+b^2+c^2)
   • Dot Product
     • if a is perpendicular to b, then the dot product is 0 -any vector dotted with itself is 1
     • Geometric Definition
       • a . b= ||a||*||b||*cos(ab)
     • Algebraic Definition
       • a1b1+a2b2+a3b3
   • Cross Product
     • Geometric Definition
       • axb= ||a||*||b||*sin(theta)
     • Algebraic Definition
       • (a2b3-a3b2)i+(a3b1-a1b3)j+(a1b2-a2b1)k
     • Matrix
       • to find i: block row and column w/ i, you get a smaller matrix(abcd) (ad-bc)i
   • Vector Fields
     • Important Vector Fields
       • xi+yj looks like an explosion -yi+xj swirling counter clockwise

6. Parameterization

   • Finding Parametric Equations
     • Given a vector and a point: -translate the point into a vector -set up an equation where you add the new vector and the old vector multiplied by t   -Factor
     • Example
       • V=i+2j-k     (3,0,-4) (3,0,-4)-->3i-4k r=3i-4k+t(i+2j-k) r=(3+t)i+(2t)j-(4-t)k x=3+t y=2t z=-4-t