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Magnitude of a vector

‖v‖ = √(x² + y² + z²)
Use it to find vector length.

Unit vector

u = v / ‖v‖
Use when you need direction only (length = 1).

Dot product

A · B = AxBx + AyBy + AzBz = ‖A‖‖B‖cos(θ)

Perpendicular vectors

A · B = 0

Angle between vectors

cos(θ) = (A · B)/(‖A‖‖B‖)

Cross product

A × B gives a vector perpendicular to both A and B.

Area of parallelogram

Area = ‖A × B‖

Line in R³

"(x

Plane from point + normal

a(x−x₀) + b(y−y₀) + c(z−z₀) = 0

Distance point to plane

d = |ax₀ + by₀ + cz₀ + d| / √(a²+b²+c²)

Angle between planes

Use normals:
cos(θ) = |n₁·n₂|/(‖n₁‖‖n₂‖)

Square root domain rule

Radicand ≥ 0

Logarithm domain rule

Argument > 0

arcsin domain rule

−1 ≤ value ≤ 1

Sphere

x² + y² + z² = r²

Ellipsoid

x²/a² + y²/b² + z²/c² = 1

Cone

x²/a² + y²/b² − z²/c² = 0

Hyperboloid of one sheet

x²/a² + y²/b² − z²/c² = 1

Elliptic paraboloid

x²/a² + y²/b² = z/c

Electric field (point charge)

E = kₑ q / r² · r̂

Solve: line through two points

"1. Compute direction vector: v = P₂ − P₁<br>2. Use one point P₀<br>3. Write: (x

Solve: plane through 3 points

1. Build PQ and PR<br>2. Compute normal: n = PQ × PR<br>3. Use point-normal form

Solve: distance point → plane

"1. Identify a

Solve: angle between planes

1. Extract normals n₁ and n₂<br>2. Use dot product formula<br>3. Solve for θ

Solve: plane perpendicular to two planes

1. Get normals n₁ and n₂<br>2. Compute n = n₁ × n₂<br>3. Use given point

Solve: domain with square root

Force radicand ≥ 0<br>Also denominator ≠ 0 if needed<br>Solve inequalities

Solve: domain with logarithm

Force argument > 0<br>For ln(ln(...)): inner > 0 and ln(...) > 0

Solve: domain with arcsin

Force −1 ≤ expression ≤ 1<br>Combine with denominator restrictions

Solve: electric field

1. Vector from each charge to P<br>2. Compute distance r<br>3. Apply E = kₑq/r² · r̂<br>4. Add vectors

Solve: boat + current

1. Write both velocity vectors<br>2. Add vectors<br>3. Magnitude = speed<br>4. arctan = direction