MATH1131 Chap.1 Vectors, Lines and Planes
Vectors, Lines and Planes
This chapter sets up the language of the algebra strand. A vector is read in two ways at once: as an arrow with length and direction, and as an ordered list of real numbers in R^n. Lines are then written in parametric vector form, a fixed point plus every real multiple of a direction vector, which works identically in two, three or more dimensions.
Planes need two non-parallel directions, and through the origin a plane is the span of those two vectors. In three dimensions, eliminating the two parameters leaves a single Cartesian equation, which is the most efficient way to test whether a point lies on a plane.
The chapter also covers converting between parametric and Cartesian descriptions of lines, testing whether a point is on a line by finding one parameter value that satisfies every coordinate, and deciding whether two lines in space intersect, are parallel, or are skew. Skew lines cannot occur in the plane, so this is the first place where intuition from school geometry needs checking against algebra.
What this chapter covers
- 01
Vectors as arrows and as elements of R^n
- 02
Addition, scalar multiples and position vectors
- 03
Parametric vector form of a line
- 04
Symmetric Cartesian form of a line in space
- 05
Linear combinations and span
- 06
Parametric and Cartesian equations of a plane
- 07
Intersecting, parallel and skew lines
Worked example · free
Parametric line from two points and a membership test
- 1Direction: Q - P = (3, -2, 4).
- 1Line: x = (2, 1, -1) + t(3, -2, 4), so x = 2 + 3t, y = 1 - 2t, z = -1 + 4t.
- 1For R, the x-coordinate gives 2 + 3t = 11, so t = 3.
- 1Check the other coordinates: y = 1 - 6 = -5 and z = -1 + 12 = 11, both matching, so R is on the line.
Key terms
- Position vector
- The vector from the origin to a point, used to locate that point algebraically.
- Direction vector
- Any non-zero vector parallel to a line, used as the multiple added to a fixed point.
- Span
- The set of all linear combinations of the given vectors.
- Skew lines
- Lines in space that are neither parallel nor intersecting.
- Parametric vector form
- A description of a line or plane as a fixed point plus real multiples of direction vectors.
Vectors, Lines and Planes FAQ
Why does the course prefer parametric form to y = mx + c?
The parametric form describes lines in any number of dimensions and handles vertical lines without special cases, while slope-intercept form only describes non-vertical lines in the plane.
Can two different-looking parametric equations describe the same line?
Yes. Any point on the line and any non-zero parallel direction give a valid description. Check equivalence by substituting one point into the other equation and confirming the directions are scalar multiples.
How do I tell whether lines in space are skew?
Equate the coordinates and try to solve for both parameters. If no solution exists and the direction vectors are not multiples of each other, the lines never meet and are not parallel, which is exactly what skew means.
How do I get a Cartesian equation of a plane from three points?
Form two direction vectors from the points, write the parametric form, then eliminate both parameters, or take a normal vector from the cross product in the next chapter. Verify the final equation with all three points.
What is the difference between a point and a position vector?
A point is a location, while its position vector is the arrow from the origin to that location. Writing points as position vectors lets you add directions to them and so describe lines and planes algebraically.
Exam move
Practise translating each object between its two descriptions until it is automatic: a line from two points into parametric form and back to symmetric Cartesian form, and a plane from three points into a single Cartesian equation. After every answer, substitute the given points to check it.
Sketch a rough diagram for each problem in three dimensions, even an imperfect one, because it reveals whether an answer such as skew or parallel is plausible. Use the recommended homework problems from the Algebra Notes for volume, and the Möbius lessons for quick feedback on routine conversions.
When a question gives a line in symmetric form, convert it to parametric form before doing anything else, because intersection and membership tests are simpler with explicit coordinates in one parameter.
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