Vectors and the Geometry of Space

The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth ($x$, $y$, and $z$).

Vectors are needed in order to describe a plane and can give the direction of all dimensions in one vector equation.

A Euclidean vector is a geometric object that has magnitude (i.e. length) and direction.

The dot product takes two vectors of the same dimension and returns a single value.

The cross product of two vectors is a vector which is perpendicular to both of the original vectors.

A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies.

A quadric surface is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial.

Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.

A surface is a two-dimensional, topological manifold.