MTH2010 Chap.1 Vectors, Lines, Planes and Curves in Space
Vectors, Lines, Planes and Curves in Space
Week 1 of the unit sets up the geometry that every later topic relies on. A point in three-dimensional space is identified with its position vector, and lines, planes and curves are all described by a base point together with direction vectors. Two operations carry most of the computation.
The dot product gives the angle between vectors and detects orthogonality, while the cross product of two vectors produces a third vector perpendicular to both, which is how a normal to a plane is found from two directions lying in it.
Lines appear in parametric form, as a point plus a multiple of a direction, and in symmetric form, where the ratios of coordinate differences to direction components are set equal and any zero component is simply left out. Planes are written either as a point with a normal, giving an equation of the form n1(x - x0) + n2(y - y0) + n3(z - z0) = 0, or parametrically from one point and two independent directions.
Curves in space are traced by vector-valued functions r(t), differentiated component by component, and the unit assumes the derivative never vanishes so that each point has a tangent direction.
Typical questions ask for the plane through three points and whether another line meets it, for the tangent line to a curve at a named point, for the angle at which two curves cross, or for the points where a tangent is parallel to a given vector.
What this chapter covers
- 01
Dot product: angle between vectors and the test for orthogonality
- 02
Cross product: a vector perpendicular to two given directions, and parallelepiped volume
- 03
Lines: parametric form through two points and symmetric form with zero components omitted
- 04
Planes: point and normal form, parametric form, and conversion between them
- 05
Line against plane: direction dotted with the normal, then one point substituted
- 06
Parametric curves: componentwise derivatives and the regularity assumption
- 07
Tangent lines: find the parameter value first, then use r(t0) and r'(t0)
- 08
Curve intersections: separate parameters, meeting point, angle between tangents
Worked example · free
Plane through three points
- 1Edge vectors: PQ = (2, 0, 1) and PR = (0, 2, 2).
- 1Normal: PQ x PR = (0*2 - 1*2, 1*0 - 2*2, 2*2 - 0*0) = (-2, -4, 4), a multiple of (1, 2, -2).
- 1Point-normal form through P: (x - 1) + 2(y - 2) - 2z = 0, so x + 2y - 2z = 5.
- 1Check: Q gives 3 + 4 - 2 = 5 and R gives 1 + 8 - 4 = 5.
Key terms
- Position vector
- The vector from the origin to a point, used interchangeably with the point itself.
- Normal vector
- A non-zero vector perpendicular to every direction lying in a plane.
- Cross product
- An operation on two vectors in three dimensions that returns a vector perpendicular to both, with length equal to the area of their parallelogram.
- Regular parametrisation
- A parametrisation of a curve whose derivative is never the zero vector.
- Tangent line
- The line through a point of a curve in the direction of the derivative of the parametrisation there.
Vectors, Lines, Planes and Curves in Space FAQ
How do I find a plane through three points?
Form two edge vectors from one of the points, take their cross product to get a normal, then write the point-normal equation. Check the answer by substituting all three original points.
How can I tell if a line is parallel to a plane?
Dot the direction of the line with the normal of the plane. A zero result means parallel; then test one point of the line in the plane equation to separate lying inside from never meeting.
Why does reparametrising a curve change its tangent vector?
Changing the parameter changes the speed at which the curve is traced, which rescales the derivative. The direction of the tangent, and therefore the tangent line itself, stays the same as long as the speed factor is non-zero.
What does it mean for two curves to intersect orthogonally?
At the meeting point their tangent vectors have zero dot product. Find the meeting parameters separately for each curve, differentiate both, evaluate at those parameters and compute the dot product.
Exam move
Practise switching every object between its two descriptions: a line in parametric and symmetric form, a plane as a normal equation and as a parametrisation. Then drill the three-point plane routine until the cross product is automatic, checking each answer by substitution.
For curves, write the parameter value on its own line before differentiating, because the first seminar quiz and the opening mid-semester questions gave separate marks for identifying it. Finish by working a tangent-parallel-to-a-vector problem, where trigonometric components produce a whole family of solutions, and state the general solution clearly.
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