Reading...
Play button
Play button
Use LEFT and RIGHT arrow keys to navigate between flashcards;
Use UP and DOWN arrow keys to flip the card;
H to show hint;
A reads text to speech;
23 Cards in this Set
- Front
- Back
|
Sphere equation standard form
|
x^2 + y^2 + z^2 = r^2
^centered at origin |
|
|
Magnitude of vector [x, y]
|
sqrt(x^2 + y^2)
|
|
|
Direction of vector [x, y]
|
arctan( y / x )
|
|
|
Adding vectors
|
Sum each component
|
|
|
Scaling a vector (in general)
|
Multiply each component by the scalor
|
|
|
Scaling a vector to unit length
|
Divide each component by the vector magnitude
|
|
|
"direction cosines"
x = ? |
x = r*cos(alpha)
where r is the vector magnitude and alpha is the angle in the x direction |
|
|
"direction cosines"
y = ? |
y = r*cos(beta)
where r is the vector magnitude and beta is the angle in the y direction |
|
|
"direction cosines"
z = ? |
z = r*cos(gamma)
where r is the vector magnitude and gamma is the angle in the z direction |
|
|
A dot B =
|
|A|*|B|*cos(theta)
or multiply each component, sum the results (not a vector result) |
|
|
How to find angle between vectors A and B using dot product?
|
cos(theta) = A dot B / |A|*|B|
|
|
|
A dot A =
|
A dot A = |A|^2
|
|
|
If A dot B = zero then
|
A and B are perpendicular (orthogonal)
|
|
|
Work =
|
Force * distance = F dot d
|
|
|
| A x B | =
|
|A|*|B|*sin(theta)
|
|
|
How to find cross product?
|
Use matrix, results in a vector perpendicular to originals
|
|
|
Torque =
|
r x Force applied (cross product)
|
|
|
Scalar equation of a plane
|
a(x-xo) + b(y-yo) + c(z-zo) = 0
where <a,b,c> is the normal vector and (xo, yo, zo) is a given point in the plane |
|
|
Vector equation of a line
|
L(t) = <x, y, z> + <v1, v2, v3>t
<x,y,z> is location of a point on the line <v1,v2,v3> is the velocity (slope) vector of the line |
|
|
What is needed to describe a plane?
|
A normal vector and a point in the plane
|
|
|
How to tell if two planes are parallel, perpendicular, or neither?
|
Find angle between normal vectors (using dot product and inverse cosine):
theta = 0: perpendicular theta = 1 or -1: parallel other: neither |
|
|
How to find line of intersection of two planes?
|
Cross normal vectors to get the line's velocity vector, find a point on the line by letting one parameter be zero and solve the system of the two remaining equations
|
|
|
How to find if/where two lines intersect?
|
Parameterize for L1 and L2 using t = t1 and t2. Set the x equations equal and the y equations equal, solve for t1 and t2, and then check the z coord of both lines to see if they are equal
|
