Formulas

Distance between two points:
AB = B − A =

(xB − x A )2 + ( yB − y A )2 + (zB − z A )2

Dot product (inner product) of two vectors:
u ⋅ v = u1v1 + u2v2 + u3v3 = u v cos ϕ
Angle between straight lines:

ϕ=

u ⋅v
u v

Vector product (cross product) of two vectors:
i
j k

u × v = u1 u2
v1

v2

u3 = (u2v3 − u3v2 , − u1v3 + u3v1 , u1v2 − u2v1 )
v3

Value of vector product:

u × v = u v sin ϕ

Conic sections

circle x 2 + y 2 = r 2

ellipse

x2 y2
+
=1
a 2 b2

circle ( x − m ) + ( y − n ) = r 2
2

ellipse

2

x2 y 2
+
=1
a 2 b2

x2 y2
hyperbola 2 − 2 = 1
a
b

equiaxed hyperbola y =

(

k
, u= k
x

x2 y2
hyperbola − 2 + 2 = 1
a
b

)

(

k
equiaxed hyperbola y = − , u = k
x

parabola x 2 = 2 py

parabola x 2 = −2 py

parabola y 2 = 2 px

parabola y 2 = −2 px

)

Quadrics

sphere x 2 + y 2 + z 2 = r 2

hyperboloid of one sheet

cone

x2 y2 z 2
+
−
=1
a 2 b2 c2

x2 y2 z 2
+
−
=0
a 2 b2 c 2

ellipsoid

x2 y 2 z 2
+
+
=1
a 2 b2 c 2

hyperboloid of two sheets

elliptic paraboloid

x2 y2 z 2
−
−
=1
a2 b2 c2

x2 y2
+
=z
a2 b2

hyperbolic paraboloid

elliptic cylinder

x2 y 2
−
=z
a 2 b2

x2 y2
+
=1
a 2 b2

hyperbolic cylinder

x2 y2
−
=1
a 2 b2

hyperbolic paraboloid (rotated) z = axy

x2
=y
a2