PHYS2150 Chap.1 Three-Dimensional Coordinates and Surfaces
Three-Dimensional Coordinates and Surfaces
Geometry before algebra
Three-dimensional coordinate geometry becomes systematic once every object is represented by the information that defines it. A point needs three coordinates. A line needs one point and a direction vector. A plane can be specified by a point and a normal vector.
Writing these ingredients before manipulating equations keeps the geometry visible: dot products test perpendicularity, cross products construct common normals, and substitution decides incidence. Parametric, symmetric and vector forms describe the same line, but they serve different purposes.
Parametric form is convenient for intersections, symmetric form exposes direction ratios, and vector form makes projection calculations compact.
Lines, planes and distances
A direction test must come before a distance formula. For a line and plane, compare the line direction with the plane normal.
A nonzero dot product means the line crosses the infinite plane, so the correct task is to solve for the intersection parameter. A zero dot product means the line is parallel to the plane, but one more test is required: substituting a point from the line reveals whether the line lies in the plane or remains separate. Point-to-plane distance is an absolute projection onto a unit normal.
Distances involving skew lines use the common normal obtained from the cross product of their directions.
Recognising surfaces from traces
An equation in three coordinates represents a surface, and its two-dimensional traces often reveal the surface more clearly than a perspective sketch. Fix one coordinate and identify the resulting curve, then repeat for the other coordinate planes.
A sphere gives circular traces that shrink toward its poles. A cylinder repeats one curve along its axis. Ellipsoids rescale spherical geometry, while cones, paraboloids and hyperboloids are distinguished by powers and sign patterns. The axis is not guessed from the page layout; it follows from the coordinate whose role differs from the other two.
A trace table at several fixed values is usually enough to verify the classification.
Coordinate changes follow symmetry
Cylindrical and spherical coordinates are alternatives to Cartesian coordinates, not decorations added after a solution. Cylindrical coordinates preserve height while replacing the horizontal plane by radius and angle, which suits surfaces or regions with an axis of symmetry.
Spherical coordinates use radial distance and two angles, which suits spheres, cones and genuinely radial boundaries. The best system is the one that makes the complete region easier to describe.
A single boundary may look simple in one system while another boundary becomes awkward, so check every limit before committing to the transformation.
Checks that expose geometric errors
Verification should return to the object rather than repeat the algebra. Substitute an intersection point into every defining equation. Confirm that a direction vector is perpendicular to the claimed normal.
Check that a unit vector has magnitude one and that every distance is nonnegative. For a classified surface, inspect a central trace and a trace away from the origin to see whether radii grow, shrink or split into separate pieces. These short checks distinguish a correct geometric branch from a polished calculation based on the wrong relationship.
What this chapter covers
- 01
Vector and parametric line equations
- 02
Plane normals and incidence
- 03
Distances between geometric objects
- 04
Quadric surfaces from traces
- 05
Cylindrical and spherical coordinates
- 06
Intersection and substitution checks
Worked example · free
Line against a plane
- 2Compare direction with normal.
- 2Test incidence or solve the parameter.
- 2Interpret the geometric branch.
Key terms
- Normal Vector
- A vector perpendicular to a line, plane or surface at the point considered.
- Skew Lines
- Nonparallel lines in space that do not intersect.
- Direction Vector
- A nonzero vector parallel to a line that determines the line's orientation.
- Surface Trace
- A curve formed by intersecting a surface with a coordinate plane or another fixed-coordinate plane.
- Cylindrical Coordinates
- A radius, angle and height description adapted to geometry around an axis.
- Spherical Coordinates
- A radial distance and two angles used to describe points with radial symmetry.
Three-Dimensional Coordinates and Surfaces FAQ
When is a line parallel to a plane?
Its direction vector is perpendicular to the plane normal, so their dot product is zero. That condition alone does not distinguish a separate parallel line from one lying in the plane. Substitute one line point into the plane equation before deciding whether a nonzero distance exists.
How should a quadric surface be classified?
Inspect the powers and sign pattern, then take traces by fixing one coordinate at several values. Circles, ellipses, hyperbolas or parabolas reveal the axis and whether the surface has a waist, vertex or separate sheets. Check the central trace before relying on a memorised name.
When are spherical coordinates useful?
They are effective when radial distance and two angles describe every boundary more simply than Cartesian coordinates, especially for spheres, cones and radially symmetric regions. Test all boundary equations and angle ranges first, because simplifying one surface does not guarantee that the complete region becomes easier.
How is the distance between skew lines organised?
Take the cross product of the two direction vectors to obtain a common normal, then project a displacement between chosen points onto its unit form. Before using that formula, verify that the directions are not parallel and that the simultaneous line equations have no intersection.
Exam move
Begin each line or plane problem with a four-line geometry ledger: known point, direction, normal and requested relationship. Compute the decisive dot or cross product before selecting a formula. After solving an intersection, substitute the point into every original equation; after finding a distance, check nonnegativity and identify the unit normal used.
For surface questions, build a trace table with one central slice and at least two displaced slices, then name the axis and explain whether the traces grow, shrink or split. Redraw the same surface in Cartesian, cylindrical or spherical variables only after its symmetry is clear.
During revision, mix classification questions with incidence and distance questions so that recognising an object does not become detached from using its equations. Keep a short error log for wrong branches, especially treating parallel as coincident, forgetting to normalise a projection vector, or changing coordinate systems before all boundaries have been inspected.
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