intersects the axis line, the surface of revolution is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder. Quadric surfaces In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, where and are real numbers and not all of and are zero, is called a quadric surface. There are six types of non-degenerate quadric surfaces: Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloid The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane and all the lines of through that conic that are normal to ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus. In linear algebra Another way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors. Dot product, angle, and length A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors and is defined as: The magnitude of a vector is denoted by . The dot product of a vector with itself is which gives the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors and is given by where