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Analytic geometry

Point: ( , , )o o oT x y z

Distance between 2 points: 2 2 21 1 0 1 0( ) ( ) ( )ox x y y z z

Directional vector is a vector with start point = (0,0,0): , ,x y zr a a a or ( , , )x y zr a a a

Important: ( , , )x y zr a a a has same direction as ANY MULTIPLE ( , , )x y zt r t a t a t a

Ex: ( 3,2,0)r is same direction as (6, 4,0)r (multiplied by any number; here -2)

Vector AB between 2 points ( , , )x y zA a a a and ( , , )x y zB b b b has

directional vector: ( , , )AB x x y y z zr b a b a b a subtract in same direction!

Length (Intensity, Module) of vector: 2 2 2| | ( ) ( ) ( )x y zr a a ar

Unit directional vector for r is ( , , )yx zu

aa arr r r

or minus this (divide each component by length)

Ex: 3 2( , ,0) ( 0,83; 0,55; 0)13 13ur or 3 2( , ,0)

13 13ur (multiply by -1)

Line: 3 forms (sometimes greek letter for parameter instead of t and no form is unique!)

p: o o o

x y z

x x y y z za a a

parametric 1: ( , , ) ( , , )o o o x y zp x y z t a a a

parametric 2: o x

o y

o z

x x t ay y t a

z z t a

Equation of line through point ( , , )o o oT x y z and parallel to ( , , )x y zr a a a

Ex: (1,0, 3)T and parallel to (1,2,0)r is (1,0, 3) (1,2,0)p t or 1 31 2

x y z

Equation of line through 2 points ( , , )o o oT x y z and 1 1 1( , , )P x y z

Form parallel directional vector: 1 1 0 1 0, , )(TP or x x y y z z and use either point.

Plane: 1 form (not unique)

0Ax By Cz D

Equation of plane through point ( , , )o o oT x y z and normal to ( , , )x y zr a a a

( ) ( ) ( ) 0x o y o z oa x x a y y a z z

Ex: (1,0, 3)T and normal to (1,2,0)r is: 1( 1) 2( 0) 0( 3) 0x y z or 2 1x y

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Equation of plane through 3 points: Form normal vector by first finding 2 parallel directional vectors and

taking their vector product and use any point.

Equation of plane through (a,0,0), (0,b,0) and (0,0,c): 1x y za b c

Angles: Remember – use the parallel and normal directional vectors of lines and planes! The directional vector normal to two directional vectors is their vector product. Directional vectors are normal if scalar product is 0.

Give 2 directional vectors: ( , , )x y za a a a and ( , , )x y zb b b b

a b 0a b 0x x y y z za b a b a b

Angle between: 2 2 2 2 2 2

cos ( , )| | | | ( ) ( ) ( ) ( ) ( ) ( )

x x y y z z

x y z x y z

a b a b a ba ba ba b a a a b b b

Normal to both vectors: ab x y z

x y z

i j kn a b a a a

b b b

Condition 1 is proof; condition 2 is use.

a b 0,0,0x y z

x y z

i j ka b a a a

b b b

| | | |a b a b

Result: Two planes are parallel if their normals are parallel.

a R 0,0,0x y z

x y z

i j ka n a a a

n n n

| | | |a n a n

Point intersection of line and plane. ( , , ) ( , , )o o o x y zp x y z t a a a 0Ax By Cz D .

Solve for t: ( ) ( ) ( ) 0o x o y o zA x t a B y t a C z t a D and substitute back into line.

Line intersection of planes. 0Ax By Cz D and 1 1 1 1 0A x B y C z D

Find normal of normals (parallel to both planes) and find intersection point (pick x0=1 and find yo and zo).

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Other formulas

Distance between 2 planes: 0Ax By Cz D and 1 0Ax By Cz D is 1| |D D .

Distance between 2 lines: ( , , ) ( , , )o o o x y zp x y z t a a a and 1 1 1( , , ) ( , , )x y zq x y z t b b b

1. Find normal between parallels: ab x y z

x y z

i j kn a b a a a

b b b

. If <0,0,0> then lines are parallel.

2. Find intensity | |abn .

3. Find a point on each line: ( , , )o o oA x y z and 1 1 1( , , )B x y z

4. Distance = | |

ab

ab

nB An

(scalar product)

If lines are parallel: | |

B A aa

See: http://youtube.com/lfahlberg Playlist: 3D Geometry