formulas
• Equation of line in vector parametrization form
• Projection of u onto v
r(t) = hx0 , y0 , z0 i + tha, b, ci
projv u =
• Equation of line: parametric equation
x = x0 + at, y = y0 + bt, z = z0 + ct
• Magnitude of v = ha, b, ci
p
kvk = a2 + b2 + c2
u · v
v·v
v
• Equation for tangent plane to z = f (x, y) at (a, b)
z = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)
• Length of path r(t) = hx(t), y(t), z(t)i
• Dot product and the Angle
x · y = kxkkyk cos θ
• The cross product of v = ha1 , b1 , c1 i and v =
ha2 , b2 , c2 i:
Z
b
kr0 (t)kdt =
p
x0 (t)2 + y 0 (t)2 + z 0 (t)2
a
• Product rule for vector-valued functions.
v×w = (b1 c2 −b2 c1 )i−(a1 c2 −c1 a2 )j+(a1 b2 −a2 b1 )k
• Area of parallelogram spanned by v and w
d
(r1 (t)r2 (t)) = r01 (t)r2 (t) + r1 (t)r02 (t)
dt
kv × wk = kvkkwk | sin θ|
• Cross product rule for vector-valued functions.
where θ is angle between v and w.
• Volume of parallelepiped spanned by u, v and w.
|u · (v × w)|
d
(r1 (t) × r2 (t)) = [r1 (t) × r02 (t)] + [r1 (t) × r02 (t)]
dt
• Equation of the plane through P0 = (x0 , y0 , z0 ) • Tangent line at r(t0 )
with normal vector n = ha, b, ci.
L(t) = r(t0 ) + t r0 (t0 )
Vector form
Scalar form
n · hx, y, zi = d
ax + by + cz = d
• Gradient of f (x, y)
where d = n · hx0 , y0 , z0 i
∇f (x, y) = hfx , fy i
• Cylindrical Coordinates (r, θ, z)
Cylind. to Rect.
x = r cos θ
y = r sin θ
z = z
Rect. to Cylind.
p
r =
x2 + y 2
y
tan θ =
x
z = z
• Spherical Coordinates (ρ, θ, φ)
Spherical to Rect.
x = ρ cos θ sin φ
y = ρ sin θ sin φ
z = ρ cos φ
• Chain rule for composition of function f (x, y) and
path c(t) = hx(t), y(t)i
d
(f (c(t))) = ∇fc(t) · c0 (t)
dt
• Discriminant
D = fxx (a, b)fyy (a, b) − fxy (a, b)2
Rect. to Spherical
p
ρ =
x2 + y 2 + z 2 • Equation of tangent plane to surface F (x, y, z) = k
y
at P = (a, b, c)
tan θ =
x
z
cos φ =
Fx (a, b, c)(x−a)+Fy (a, b, c)(y−b)+Fz (a, b, c)(z−c) = 0
ρ