Unit 1 - created from Mind Map

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Navigating 3-Space<br><br><ul><li>going from (x,y) to (x,y,z)</li></ul>

Linear Functions

Cross Sections

Contour Diagrams<br><br><ul><li>-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</li></ul>

Vectors

Parameterization

Right Hand Rule<br><br><ul><li>Put your index finger along the x axis, middle finger along the y axis, and your thumb will always point in the z direction</li></ul>

Important Objects in 3 Dimensions<br><br><ul><li>xy plane: z=0

xz plane: y=0

yz plane: x=0</li></ul>

Z=mx+ny+c

Finding the Equation of a Linear Function<br><br><ul><li>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.</li></ul>

Recognizing Linear Functions

m= Dz/Dx n=Dz/Dy

Cross Sections<br><br><ul><li>all equally spaced, parallel lines</li></ul>

Contour Diagrams<br><br><ul><li>Parallel lines</li></ul>

Tables<br><br><ul><li>check and make sure all changes in the x,y,z directions are constant or are approx. constant</li></ul>

In Y<br><br><ul><li>-set x equal to a constant

-create a table to find the equations of the cross sections

-graph the functions </li></ul>

In X<br><br><ul><li>-set y equal to a constant

-create a table to find the -equations of the cross sections

graph the functions </li></ul>

Finding the Length of a Vector<br><br><ul><li>for the vector v=ai+bj+ck

||v||= sqrt(a^2+b^2+c^2)</li></ul>

Dot Product<br><br><ul><li>if a is perpendicular to b, then the dot product is 0

-any vector dotted with itself is 1</li></ul>

Cross Product

Vector Fields

Geometric Definition<br><br><ul><li>a . b= ||a||*||b||*cos(ab)</li></ul>

Algebraic Definition<br><br><ul><li>a1b1+a2b2+a3b3</li></ul>

Geometric Definition<br><br><ul><li>axb= ||a||*||b||*sin(theta)</li></ul>

Algebraic Definition<br><br><ul><li>(a2b3-a3b2)i+(a3b1-a1b3)j+(a1b2-a2b1)k</li></ul>

Matrix<br><br><ul><li>to find i:

block row and column w/ i, you get a smaller matrix(abcd)

(ad-bc)i</li></ul>

Important Vector Fields<br><br><ul><li>xi+yj

looks like an explosion

-yi+xj

swirling counter clockwise</li></ul>

Finding Parametric Equations<br><br><ul><li>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</li></ul>

Example<br><br><ul><li>V=i+2j-k (3,0,-4)

(3,0,-4)--&amp;gt;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</li></ul>