Equations Flashcards
R3 vector equation - plane
(X, y, z) = (x0, y0, z0) + s(a1, a2, a3) + t(b1, b2, b3), s,tER
R3 parametric equation - plane
x = x0 + sa1 + tb1 y = y0 + sa2 + tb2 z = z= + sa3 + tb3
S, t ER
R3 Cartesian equation - plane
Ax + By + Cz + D = 0
D = -Ax0 - By0 - Cz0
A(x-x0) + B(y-y0) + C(z-z0) = 0
S, t ER
R2 vector equation - line
(x, y) = t(a, b) + (x0, y0)
tER
R2 parametric equation - line
x = ta + x0 y = tb + y0
TER
R2 Cartesian equation - line
Ax + By + C = 0
Ax + By -Ax0 - By0 = 0
R3 vector equation - line
(x, y, z) = t(a, b, c) + (x0, y0, z0)
tER
R3 parametric equation - line
x = ta + x0
y = tb + y0
z = tc + z0
tER
Symmetric equation
t = (x-x0)/a = (y-y0)/b = (z-z0)/c
a, b, c == 0
Only R3 lines
Angle between planes
θ = arccos (n1•n2/||n1|| ||n2||)
Distance between planes
d = |Ax + By + Cz + D| / root(A^2 + B^2 + C^2)
Angle between lines
θ = arccos (n1•n2/||n1|| ||n2||)
Or
= arccos (d1•d2/||d1|| ||d2||)
Distance between lines
d = |PoP•n|/||n||
Or
= |Ax + By + C|/||n||
Angle of inclination
arctan(y/x)
Dot product (geometric)
v•w = ||v|| ||w|| cosθ
Dot product (Cartesian)
v•w = x1x2 + y1y2 + z1z2
Unit vector formula
Unit vector = 1/magnitude (vector)
Scalar property - dot product
V•w is a real number
Commutative property - dot product
V•w = w•v
Scalar 0 property - dot product
V•0 vector = 0 = 0 vector•v
Dot product v•v
= ||v|| squared
Associativity - dot product
(Kv) • w = k(v•w) = v•(kw) for scalar k
Distributive property - dot product
U•(v+/-w) = u•v +/- u•w
Projection of a onto b
a1 = a⬇️b
||a⬇️b|| = |a•b|/||b||
a⬇️b = a•b/||b||^2 •b
u x u
= zero vector
If u x v = 0
V = ku for scalar k
u x v
= -(v x u)
NO COMMUTATIVITY
Associativity with scalar - cross product
ku x v = k(u x v) = u x (kv)
Distribution - cross product
u x (v x w) = (u x v) + (u x w)
Associativity - cross product
(v x w) x u = (v x u) + (w x u)
Cross product - geometric vectors
||u x v|| = ||u|| ||v|| sinθ
Also area of parallelogram determined by u and v
Direction of geometric vector cross product
Alphabetically - direction up
Crossed reverse alphabetically - direction down
Normal line
n = (A, B) such that PoP•n = 0
