Module 1 Flashcards
Distance of a Sphere from P to Q
|P-Q| = sqrt [ (a2-a1)^2 + (b2-b1)^2 + (c2-c1)^2 ]
SPhere of radius R centered at (a, b, c)
dist^2 = R^2
(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2
Definition of a line in 3-D
vector r = < x(t), y(t), z(t) >
where x(t), y(t), and z(t) are linear functions
If you find the equation of a line with one point and then do the same process with a different point, you will get:
a different parameterization of the same line
r(t) =
= tv = t< a, b, c > = < at, bt, ct >
How to find the vector parametric equations for a line?
Use the two points P and Q to find PQ (Q-P). That vector is then the vector parameteriaztion and one of the points P or Q is the point vector.
perpendicular to the xz plane:
< 0, 1, 0 >
perpendicular to the xy plane:
< 0, 0, 1 >
perpendicular to the yz plane:
< 1, 0, 0 >
How do you find if two lines intersect?
set one variable of t equal to s. set up two systems of equations and set two variables equal to each other. find t and s. plug t and s into the last variable equal equation.
if the final equations do not match the lines do not intersect.
If they do intersect, plug in t or s into the corresponding r1 or r2 equation to find the point.
Dot Product
V . W
= a1a2 + b1b2 + c1c2
V . V
||V||^2 = || < a, b, c > || = sqrt(a^2 + b^2 + c^2)
cos (pheta)
= ( V . W) / (||V|| ||W||)
to find the angle: multipluy this my the inverse cosine
two non-zero vectors are called:
normal/ perpendicular/ orthongonal
acute angle =
V . W > 0
cos (pheta) > 0
obtuse angle =
V . W < 0
cos (pheta) < 0
right angle =
V perpedicular to W -> V . W = 0
how to find a vector normal (perpendicular) to a plane:
a(x-x0) + b(y-y0) + c(z-z0) = 0 or ax + by + cz = d
< a, b, c > for both situations
Cross Product
< bf - ec, cd - fa, ae - db >
|abc/def| start in second column
3 x 3 Determinant
(aei + bfg + cdh) - (gec + hfa + idb)
|abc/def/ghi| start in first column all the way to the third.
unit vector in the firection of V : V / ||V|| =
vector e sub vector v
parallel to vector V =
“scalar multiple of vector V”
Projection of vector U onto vector V: U sub || V =
(U . V / V . V) (V)
V . V = ||V||^2
Component/ scalar component of U along V =
U . V / ||V||
The absolute value of this is equal to the length of the projection vector
Work (W) =
F . d
units are ft/lbs unless specified otherwise
1.
||V x W|| =
||V|| ||W|| sin (pheta)
W x V =
- (V x W)
V x V =
0
V x W = 0 if:
V and W vectors are parallel {||V|| ||W|| sin (pheta) } or one or both of them are vector 0
i =
< 1, 0, 0 >
j =
< 0, 1, 0 >
k =
< 0, 0, 1 >
to determine a cross product of i, j, and k, take the shortest parth from 1st to 2nd (i, j, k in clockwise rotation)
if clockwise: positive, if counterclockwise: negative
parallelogram spanned by V and W has area:
A = ||V x W||
parallelpiped spanned by U, V, and W has volume:
V = | U. (V X W) | = 3X3 determinant
just need absolute value. order does not matter!
to find a plane you need a:
point on the plane and a normal vector to the plane
two vectors are perpendicular if their cross product is equal to zero!!
vector form
n . < x, y, z> = d = ax0 + by0 + cz0
scalar form
a(x-x0) + b(y-y0) + c(z-z0) = 0
ax + by + cz = d (standard form)
