Quadric surfaces are the 3D graphs of second-degree equations in x, y and z, such as x²/a² + y²/b² + z²/c² = 1. The six main types are the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the elliptic cone, and the hyperboloids of one and two sheets. You identify and sketch them with traces.
The equation of a plane through the point (x₀, y₀, z₀) with normal vector n = ⟨a, b, c⟩ is a(x − x₀) + b(y − y₀) + c(z − z₀) = 0, which simplifies to ax + by + cz = d. The normal vector is perpendicular to every direction in the plane. For three points, find n with a cross product.
Equations of lines in 3D describe the line through a point (x₀, y₀, z₀) in the direction of a vector v = ⟨a, b, c⟩. The vector equation is r = r₀ + tv. The parametric equations are x = x₀ + at, y = y₀ + bt, z = z₀ + ct, and the symmetric equations are (x − x₀)/a = (y − y₀)/b = (z − z₀)/c.
Vector projections measure how much of one vector points along another. The scalar projection of b onto a is (a · b) ÷ ‖a‖, the signed length of b’s “shadow” on a. The vector projection is ((a · b) ÷ ‖a‖²) a, the shadow itself as a vector. Together they split b into parts parallel and perpendicular to a.
The cross product a × b of two 3D vectors is a new vector perpendicular to both of them. Its components are ⟨a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁⟩, which come from a 3 × 3 determinant. Its length, ‖a‖‖b‖ sin θ, equals the area of the parallelogram the two vectors span.
Vectors in 3D have three components, written ⟨a, b, c⟩ or ai + bj + ck. The magnitude (length) of ⟨a, b, c⟩ is √(a² + b² + c²). You add vectors and multiply them by scalars one component at a time, and the dot product a₁b₁ + a₂b₂ + a₃b₃ gives the angle between two vectors.
A 3D coordinate system locates every point in space with an ordered triple (x, y, z), using three perpendicular axes that meet at the origin. It extends the coordinate plane by adding a z-axis for height. With it you can plot points, find the distance between two points and write the equation of a sphere.