Module 1 Flashcards

1
Q

Distance of a Sphere from P to Q

A

|P-Q| = sqrt [ (a2-a1)^2 + (b2-b1)^2 + (c2-c1)^2 ]

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2
Q

SPhere of radius R centered at (a, b, c)

dist^2 = R^2

A

(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2

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3
Q

Definition of a line in 3-D

A

vector r = < x(t), y(t), z(t) >

where x(t), y(t), and z(t) are linear functions

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4
Q

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

a different parameterization of the same line

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5
Q

r(t) =

A

= tv = t< a, b, c > = < at, bt, ct >

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6
Q

How to find the vector parametric equations for a line?

A

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.

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7
Q

perpendicular to the xz plane:

A

< 0, 1, 0 >

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8
Q

perpendicular to the xy plane:

A

< 0, 0, 1 >

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9
Q

perpendicular to the yz plane:

A

< 1, 0, 0 >

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10
Q

How do you find if two lines intersect?

A

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.

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11
Q

Dot Product

V . W

A

= a1a2 + b1b2 + c1c2

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12
Q

V . V

A

||V||^2 = || < a, b, c > || = sqrt(a^2 + b^2 + c^2)

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13
Q

cos (pheta)

A

= ( V . W) / (||V|| ||W||)

to find the angle: multipluy this my the inverse cosine

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14
Q

two non-zero vectors are called:

A

normal/ perpendicular/ orthongonal

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15
Q

acute angle =

A

V . W > 0

cos (pheta) > 0

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16
Q

obtuse angle =

A

V . W < 0

cos (pheta) < 0

17
Q

right angle =

A

V perpedicular to W -> V . W = 0

18
Q

how to find a vector normal (perpendicular) to a plane:

A

a(x-x0) + b(y-y0) + c(z-z0) = 0 or ax + by + cz = d

< a, b, c > for both situations

19
Q

Cross Product

A

< bf - ec, cd - fa, ae - db >

|abc/def| start in second column

20
Q

3 x 3 Determinant

A

(aei + bfg + cdh) - (gec + hfa + idb)

|abc/def/ghi| start in first column all the way to the third.

21
Q

unit vector in the firection of V : V / ||V|| =

A

vector e sub vector v

22
Q

parallel to vector V =

A

“scalar multiple of vector V”

23
Q

Projection of vector U onto vector V: U sub || V =

A

(U . V / V . V) (V)

V . V = ||V||^2

24
Q

Component/ scalar component of U along V =

A

U . V / ||V||

The absolute value of this is equal to the length of the projection vector

25
Work (W) =
F . d ## Footnote units are ft/lbs unless specified otherwise
26
# 1. ||V x W|| =
||V|| ||W|| sin (pheta)
27
W x V =
- (V x W)
28
V x V =
0
29
V x W = 0 if:
V and W vectors are parallel {||V|| ||W|| sin (pheta) } or one or both of them are vector 0
30
i =
< 1, 0, 0 >
31
j =
< 0, 1, 0 >
32
k =
< 0, 0, 1 >
33
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
34
parallelogram spanned by V and W has area:
A = ||V x W||
35
parallelpiped spanned by U, V, and W has volume:
V = | U. (V X W) | = 3X3 determinant | just need absolute value. order does not matter!
36
to find a plane you need a:
point on the plane and a normal vector to the plane ## Footnote two vectors are perpendicular if their cross product is equal to zero!!
37
vector form
n . < x, y, z> = d = ax0 + by0 + cz0
38
scalar form
a(x-x0) + b(y-y0) + c(z-z0) = 0 ax + by + cz = d (standard form)