Equations

Question 1

R3 vector equation - plane

Answer 1

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

Question 2

R3 parametric equation - plane

Answer 2

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

S, t ER

Question 3

R3 Cartesian equation - plane

Answer 3

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

S, t ER

Question 4

R2 vector equation - line

Answer 4

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

tER

Question 5

R2 parametric equation - line

Answer 5

x = ta + x0
y = tb + y0

TER

Question 6

R2 Cartesian equation - line

Answer 6

Ax + By + C = 0

Ax + By -Ax0 - By0 = 0

Question 7

R3 vector equation - line

Answer 7

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

tER

Question 8

R3 parametric equation - line

Answer 8

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

Question 9

Symmetric equation

Answer 9

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

Only R3 lines

Question 10

Angle between planes

Answer 10

θ = arccos (n1•n2/||n1|| ||n2||)

Question 11

Distance between planes

Answer 11

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

Question 12

Angle between lines

Answer 12

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

Question 13

Distance between lines

Answer 13

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

Question 14

Angle of inclination

Answer 14

arctan(y/x)

Question 15

Dot product (geometric)

Answer 15

v•w = ||v|| ||w|| cosθ

Question 16

Dot product (Cartesian)

Answer 16

v•w = x1x2 + y1y2 + z1z2

Question 17

Unit vector formula

Answer 17

Unit vector = 1/magnitude (vector)

Question 18

Scalar property - dot product

Answer 18

V•w is a real number

Question 19

Commutative property - dot product

Answer 19

V•w = w•v

Question 20

Scalar 0 property - dot product

Answer 20

V•0 vector = 0 = 0 vector•v

Question 21

Dot product v•v

Answer 21

= ||v|| squared

Question 22

Associativity - dot product

Answer 22

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

Question 23

Distributive property - dot product

Answer 23

U•(v+/-w) = u•v +/- u•w

Question 24

Projection of a onto b

Answer 24

a1 = a⬇️b

||a⬇️b|| = |a•b|/||b||

a⬇️b = a•b/||b||^2 •b

Question 25

u x u

Answer 25

= zero vector

Question 26

If u x v = 0

Answer 26

V = ku for scalar k

Question 27

u x v

Answer 27

= -(v x u)

NO COMMUTATIVITY

Question 28

Associativity with scalar - cross product

Answer 28

ku x v = k(u x v) = u x (kv)

Question 29

Distribution - cross product

Answer 29

u x (v x w) = (u x v) + (u x w)

Question 30

Associativity - cross product

Answer 30

(v x w) x u = (v x u) + (w x u)

Question 31

Cross product - geometric vectors

Answer 31

||u x v|| = ||u|| ||v|| sinθ

Also area of parallelogram determined by u and v

Question 32

Direction of geometric vector cross product

Answer 32

Alphabetically - direction up

Crossed reverse alphabetically - direction down

Question 33

Normal line

Answer 33

n = (A, B) such that PoP•n = 0