Equations Flashcards

1
Q

R3 vector equation - plane

A

(X, y, z) = (x0, y0, z0) + s(a1, a2, a3) + t(b1, b2, b3), s,tER

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2
Q

R3 parametric equation - plane

A
x = x0 + sa1 + tb1
y = y0 + sa2 + tb2
z = z= + sa3 + tb3

S, t ER

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3
Q

R3 Cartesian equation - plane

A

Ax + By + Cz + D = 0
D = -Ax0 - By0 - Cz0
A(x-x0) + B(y-y0) + C(z-z0) = 0

S, t ER

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4
Q

R2 vector equation - line

A

(x, y) = t(a, b) + (x0, y0)

tER

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5
Q

R2 parametric equation - line

A
x = ta + x0
y = tb + y0

TER

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6
Q

R2 Cartesian equation - line

A

Ax + By + C = 0

Ax + By -Ax0 - By0 = 0

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7
Q

R3 vector equation - line

A

(x, y, z) = t(a, b, c) + (x0, y0, z0)

tER

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8
Q

R3 parametric equation - line

A

x = ta + x0
y = tb + y0
z = tc + z0
tER

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9
Q

Symmetric equation

A

t = (x-x0)/a = (y-y0)/b = (z-z0)/c
a, b, c == 0

Only R3 lines

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10
Q

Angle between planes

A

θ = arccos (n1•n2/||n1|| ||n2||)

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11
Q

Distance between planes

A

d = |Ax + By + Cz + D| / root(A^2 + B^2 + C^2)

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12
Q

Angle between lines

A

θ = arccos (n1•n2/||n1|| ||n2||)
Or
= arccos (d1•d2/||d1|| ||d2||)

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13
Q

Distance between lines

A

d = |PoP•n|/||n||
Or
= |Ax + By + C|/||n||

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14
Q

Angle of inclination

A

arctan(y/x)

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15
Q

Dot product (geometric)

A

v•w = ||v|| ||w|| cosθ

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16
Q

Dot product (Cartesian)

A

v•w = x1x2 + y1y2 + z1z2

17
Q

Unit vector formula

A

Unit vector = 1/magnitude (vector)

18
Q

Scalar property - dot product

A

V•w is a real number

19
Q

Commutative property - dot product

A

V•w = w•v

20
Q

Scalar 0 property - dot product

A

V•0 vector = 0 = 0 vector•v

21
Q

Dot product v•v

A

= ||v|| squared

22
Q

Associativity - dot product

A

(Kv) • w = k(v•w) = v•(kw) for scalar k

23
Q

Distributive property - dot product

A

U•(v+/-w) = u•v +/- u•w

24
Q

Projection of a onto b

A

a1 = a⬇️b

||a⬇️b|| = |a•b|/||b||

a⬇️b = a•b/||b||^2 •b

25
u x u
= zero vector
26
If u x v = 0
V = ku for scalar k
27
u x v
= -(v x u) | NO COMMUTATIVITY
28
Associativity with scalar - cross product
ku x v = k(u x v) = u x (kv)
29
Distribution - cross product
u x (v x w) = (u x v) + (u x w)
30
Associativity - cross product
(v x w) x u = (v x u) + (w x u)
31
Cross product - geometric vectors
||u x v|| = ||u|| ||v|| sinθ Also area of parallelogram determined by u and v
32
Direction of geometric vector cross product
Alphabetically - direction up | Crossed reverse alphabetically - direction down
33
Normal line
n = (A, B) such that PoP•n = 0