Analytic geometry of space: synthesis and challenges

Summary

This lesson closes the unit on the analytic geometry of space, which is the last one of the course. It gathers into a deductive chain what was achieved in its four content lessons: the distance between two points, obtained by applying the Pythagorean theorem twice; the dot product, the Cauchy-Schwarz inequality and the angle between two vectors, which coincides with that of elementary geometry; the cross product and the scalar triple product, born of the determinant of order three, read as an area, as a volume and as a coplanarity criterion; the equations of lines and planes and the classification of their relative positions by the Rouché-Capelli theorem (known in the Spanish-speaking world as the Rouché-Frobenius theorem); and, lastly, distances, angles, projections and the sphere. It explains what the unit receives from the earlier ones and from the course Algebra and Geometry I, what was admitted without proof and where it leads. It repeats in an exit assessment the skills of the entry assessment of the introduction, and it ends with ten challenges, harder than the problems of the lessons, which combine several of them.

Learning objectives

Reconstruct the chain of definitions and theorems of the unit, from the distance between two points to the common perpendicular and the sphere, pointing out in each link the earlier results that its proof uses.
Distinguish in the unit what was proved from what was admitted, and relate its results to the determinants and the systems of the unit on matrices and to the analytic geometry of the plane of Course I.
Solve problems that combine the dot, cross and scalar triple products with the equations of lines, planes and spheres, choosing in each case the most convenient tool and system of axes.
Model with the geometry of space situations from crystallography, technical drawing, optics and navigation, and communicate the solution with its setup, its computation and its plausibility check.

What the unit achieved

Coordinates and the dot product

The unit began by declaring what it was not going to prove. The lesson on vectors in space and the dot product gathered into a list the facts of Euclidean solid geometry that are admitted and, with the Pythagorean theorem applied twice, obtained the distance formula. Vectors were defined as triples, and the dot product kept the properties of the dot product of the plane, with the Cauchy-Schwarz inequality and the triangle inequality. Now, in space there is no plane figure that contains everything, and the order was reversed: the angle was defined by its cosine, which the Cauchy-Schwarz inequality places between $-1$ and $1$ , and afterwards it was proved that it measures what elementary geometry calls an angle, because the dot product depends only on the lengths of the sides of a triangle. From this followed the orthogonal projection, the direction cosines and the matrices that preserve lengths and angles.

The cross product and the scalar triple product

The search for a vector perpendicular to two given vectors led, in the lesson on the cross product and the scalar triple product, to three determinants of order two. If this is so, the properties of determinants gave at once anticommutativity and bilinearity; Lagrange’s identity gave the norm as the area of the parallelogram; and the vector triple product formula, which takes the place of associativity, proved that the vectors perpendicular to two nonparallel vectors are the multiples of their cross product. The scalar triple product turned out to be a determinant of order three: it vanishes if and only if the three vectors are coplanar, its absolute value is the volume of the parallelepiped and its square is the Gram determinant. Lastly, the lesson separated what is proved about the direction of the cross product from what is agreed upon when drawing the axes, the right-handed system, and from what is admitted, the right-hand rule.

Lines and planes

The lesson on lines and planes in space reduced membership in a plane to a single condition, $\vec{n} \cdot \vec{AP} = 0$ , with $\vec{n}$ the cross product of the direction vectors, and from it obtained the Cartesian equation, unique up to a factor. With this, the points common to several planes are the solutions of a linear system, and the Rouché-Capelli theorem received its geometric reading: the solutions of a consistent system in three unknowns form a point, a line or a plane according to the rank. From it the relative positions of planes and lines were deduced, with the scalar triple product $\vec{AB} \cdot \left( \vec{u} \times \vec{v} \right)$ as the criterion for distinguishing lines that intersect from skew lines, and the pencil of planes.

Measurements

The lesson on distances, angles and volumes in space brought the three products together with the equations. By projecting onto the normal or onto the line, it proved the formulas for the distance from a point to a plane and to a line, with the foot of the perpendicular as the only nearest point, and the central theorem of the unit: two skew lines have a single common perpendicular, whose length is a scalar triple product divided by a norm. This settles the problem of the two subway tunnels posed by the introduction. It then proved that the angles between lines and planes defined by dot products coincide with those of elementary geometry, that the area of the projection of a plane polygon is its area times the cosine of the dihedral angle, and it studied the sphere and its tangent plane.

Therefore, the unit forms a chain: the admitted facts give the distance; the distance and the properties of the dot product give the angle; determinants give the cross product and the scalar triple product; these turn membership in a plane into a linear equation; linear systems classify the positions; and orthogonal projection, applied to those equations, measures. The following figure gathers the four segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that are admitted without proof or come from earlier units or from Course I, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

AEuclidean facts of space. Admitted are the existence and uniqueness of the plane through three noncollinear points and of the plane perpendicular to a line through a point, the freedom to choose the axes and the correspondence between points and triples. Admitted without proof in this unit.
BElementary areas and volumes. Admitted are base times height for the parallelogram and the parallelepiped, half of that area for the triangle, one third of base times height for the pyramid and the additivity of area; their foundation is integral calculus. Admitted without proof in this unit.
CGeometry of the plane. The distance, the properties of the dot product, the angle between two vectors, the lemma on equal vectors and parallelograms, completing the square and the sign of the quadratic polynomial according to its discriminant, proved in Course I. Admitted without proof in this unit.
DTrigonometry. The arccosine as a bijection from $\icc{-1}{1}$ onto $\icc{0}{\pi}$ , the identity $\cos^2 \theta + \sin^2 \theta = 1$ and the law of cosines, proved in units 1 and 2. Admitted without proof in this unit.
EMatrices and determinants. The transpose of a product, the one-sided inverse, the properties of the determinant, Cramer’s rule and the equivalence between $\det A \neq 0$ and maximal rank, proved in unit 4. Admitted without proof in this unit.
FRouché-Capelli. The system $AX = B$ is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}(M)$ , and then it has as many free unknowns as the number of unknowns minus the rank; proved in unit 4. Admitted without proof in this unit.
GOrthogonal projection. If $\vec{v}$ is not zero, every vector can be written in a unique way as a multiple of $\vec{v}$ plus a vector orthogonal to $\vec{v}$ , and the square of its norm is the sum of the squares of the norms of both summands. Established in: 6. Orthogonal projection.
HPerpendiculars to two vectors. If $\vec{u}$ and $\vec{v}$ are neither zero nor parallel, the vectors perpendicular to both are exactly the multiples of $\vec{u} \times \vec{v}$ . Established in: 14. Perpendiculars to two vectors.

Segment 1 Coordinates and the dot product
- Locate points in space and compute distances, justifying the formula with the Pythagorean theorem.
- Operate with vectors in space and prove with them properties of parallelograms and parallelepipeds.
- Prove the Cauchy-Schwarz inequality and the triangle inequality, and define with them the angle between two vectors.
- Compute orthogonal projections and direction cosines, and recognize the matrices that preserve lengths and angles.
1. Distance between two points. The distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $\sqrt{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2 + \left( z_2 - z_1 \right)^2}$ , by the Pythagorean theorem applied in a horizontal plane and then in a vertical one.
  Rests on: A (Euclidean facts of space), C (Geometry of the plane). Proved in: Vectors in space and the dot product.
2. Vectors and the standard basis. Triples form a vector space, every vector is in a unique way a combination of $\vec{i}$ , $\vec{j}$ and $\vec{k}$ , and if $\vec{PQ} = \vec{RS}$ and the four points are not collinear, they lie in a plane and $PQSR$ is a parallelogram, and conversely.
  Uses: 1. Distance between two points. Rests on: A (Euclidean facts of space), C (Geometry of the plane). Proved in: Vectors in space and the dot product.
3. Properties of the dot product. The dot product is commutative and distributive, lets numbers be taken outside and satisfies $\vec{u} \cdot \vec{u} = {\norma{\vec{u}}}^2$ ; consequently, ${\norma{\vec{u} \pm \vec{v}}}^2 = {\norma{\vec{u}}}^2 \pm 2 \vec{u} \cdot \vec{v} + {\norma{\vec{v}}}^2$ .
  Uses: 2. Vectors and the standard basis. Rests on: C (Geometry of the plane). Proved in: Vectors in space and the dot product.
4. Cauchy-Schwarz and the triangle inequality. $\abs{\vec{u} \cdot \vec{v}} \le \norma{\vec{u}} \norma{\vec{v}}$ and $\norma{\vec{u} + \vec{v}} \le \norma{\vec{u}} + \norma{\vec{v}}$ hold, with equality in the first if and only if one vector is a multiple of the other.
  Uses: 3. Properties of the dot product. Rests on: C (Geometry of the plane). Proved in: Vectors in space and the dot product.
5. The angle is the geometric one. The number $\theta$ in $\icc{0}{\pi}$ such that $\cos \theta = \frac{\vec{u} \cdot \vec{v}}{\norma{\vec{u}} \norma{\vec{v}}}$ is the measure of the angle formed by the arrows of $\vec{u}$ and $\vec{v}$ drawn from the same origin; in particular, two nonzero vectors are perpendicular if and only if their dot product is zero.
  Uses: 1. Distance between two points, 3. Properties of the dot product, 4. Cauchy-Schwarz and the triangle inequality. Rests on: A (Euclidean facts of space), C (Geometry of the plane), D (Trigonometry). Proved in: Vectors in space and the dot product.
6. Orthogonal projection. If $\vec{v} \neq \vec{0}$ , every $\vec{u}$ is in a unique way $\vec{p} + \vec{w}$ , with $\vec{p} = \frac{\vec{u} \cdot \vec{v}}{{\norma{\vec{v}}}^2} \vec{v}$ and $\vec{w}$ orthogonal to $\vec{v}$ , and ${\norma{\vec{u}}}^2 = {\norma{\vec{p}}}^2 + {\norma{\vec{w}}}^2$ .
  Uses: 3. Properties of the dot product, 5. The angle is the geometric one. Proved in: Vectors in space and the dot product.
7. Orthogonal matrices. $\left( A \vec{u} \right) \cdot \vec{v} = \vec{u} \cdot \left( A^{\mathsf{T}} \vec{v} \right)$ holds; if the columns of $A$ are unit vectors and orthogonal, $A^{\mathsf{T}} A = I$ and $A$ preserves norms, distances and angles.
  Uses: 3. Properties of the dot product. Rests on: E (Matrices and determinants). Proved in: Vectors in space and the dot product.
Segment 2 The cross product and the scalar triple product
- Define the cross product by a formal determinant and prove with determinants that it is anticommutative and bilinear, but not associative.
- Prove perpendicularity, Lagrange’s identity, the parallelism criterion and the vector triple product formula.
- Identify the scalar triple product with a determinant of order three and read it as a volume and as a coplanarity criterion.
- Separate, in the direction of the cross product, what is proved from the right-handed convention and from the right-hand rule, which is admitted.
1. Algebraic properties. The cross product, whose components are three determinants of order two, is anticommutative and bilinear, satisfies $\vec{u} \times \vec{u} = \vec{0}$ and is not associative.
  Rests on: E (Matrices and determinants). Proved in: The cross product and the scalar triple product.
2. Perpendicularity. $\vec{u} \cdot \left( \vec{u} \times \vec{v} \right) = 0$ and $\vec{v} \cdot \left( \vec{u} \times \vec{v} \right) = 0$ hold: the cross product is perpendicular to its two factors.
  Uses: 3. Properties of the dot product, 8. Algebraic properties. Proved in: The cross product and the scalar triple product.
3. Lagrange’s identity. For all vectors, ${\norma{\vec{u} \times \vec{v}}}^2 = {\norma{\vec{u}}}^2 {\norma{\vec{v}}}^2 - \left( \vec{u} \cdot \vec{v} \right)^2$ holds.
  Uses: 3. Properties of the dot product, 8. Algebraic properties. Proved in: The cross product and the scalar triple product.
4. Area of the parallelogram. The norm $\norma{\vec{u} \times \vec{v}} = \norma{\vec{u}} \norma{\vec{v}} \sin \theta$ is the area of the parallelogram with sides $\vec{u}$ and $\vec{v}$ , and the triangle $ABC$ has area $\frac{1}{2} \norma{\vec{AB} \times \vec{AC}}$ .
  Uses: 5. The angle is the geometric one, 10. Lagrange’s identity. Rests on: B (Elementary areas and volumes), D (Trigonometry), G (Orthogonal projection). Proved in: The cross product and the scalar triple product.
5. Zero product and parallelism. $\vec{u} \times \vec{v} = \vec{0}$ holds if and only if one of the two vectors is a multiple of the other.
  Uses: 8. Algebraic properties. Proved in: The cross product and the scalar triple product.
6. Vector triple product. $\vec{u} \times \left( \vec{v} \times \vec{w} \right) = \left( \vec{u} \cdot \vec{w} \right) \vec{v} - \left( \vec{u} \cdot \vec{v} \right) \vec{w}$ and the Jacobi identity hold.
  Uses: 3. Properties of the dot product, 8. Algebraic properties. Proved in: The cross product and the scalar triple product.
7. Perpendiculars to two vectors. If $\vec{u}$ and $\vec{v}$ are neither zero nor parallel, a vector is perpendicular to both if and only if it is a multiple of $\vec{n} = \vec{u} \times \vec{v}$ , and then it is $\frac{\vec{x} \cdot \vec{n}}{{\norma{\vec{n}}}^2} \vec{n}$ .
  Uses: 9. Perpendicularity, 12. Zero product and parallelism, 13. Vector triple product. Proved in: The cross product and the scalar triple product.
8. The scalar triple product is a determinant. The scalar triple product $\vec{u} \cdot \left( \vec{v} \times \vec{w} \right)$ is the determinant of the matrix with rows $\vec{u}$ , $\vec{v}$ and $\vec{w}$ ; it does not change when the factors are permuted cyclically, it changes sign when two are interchanged, and the dot and the cross can be interchanged.
  Uses: 3. Properties of the dot product, 8. Algebraic properties. Rests on: E (Matrices and determinants). Proved in: The cross product and the scalar triple product.
9. Coplanarity criterion. Three vectors are coplanar, that is, they admit a zero combination with some nonzero coefficient, if and only if their scalar triple product is zero.
  Uses: 15. The scalar triple product is a determinant. Rests on: E (Matrices and determinants). Proved in: The cross product and the scalar triple product.
10. Volume and the Gram matrix. The parallelepiped with edges $\vec{u}$ , $\vec{v}$ and $\vec{w}$ has volume $\abs{\vec{u} \cdot \left( \vec{v} \times \vec{w} \right)}$ , the tetrahedron one sixth of it, and the square of the scalar triple product is the Gram determinant.
  Uses: 11. Area of the parallelogram, 12. Zero product and parallelism, 15. The scalar triple product is a determinant, 16. Coplanarity criterion. Rests on: B (Elementary areas and volumes), G (Orthogonal projection), E (Matrices and determinants). Proved in: The cross product and the scalar triple product.
11. Characterization and orientation. If $\vec{u}$ and $\vec{v}$ are neither zero nor parallel, $\vec{u} \times \vec{v}$ is the only vector perpendicular to both, with the norm of the area and with positive scalar triple product; that it points where the thumb of the right hand points is admitted, once the drawing of the axes has been agreed upon.
  Uses: 11. Area of the parallelogram, 15. The scalar triple product is a determinant. Rests on: H (Perpendiculars to two vectors). Proved in: The cross product and the scalar triple product.
Segment 3 Lines and planes
- Write lines in vector, parametric and symmetric form, and planes in parametric and Cartesian form, passing from one to another with the cross product.
- Prove that every plane has a Cartesian equation, unique up to a factor, and that every equation of the first degree describes a plane.
- Decide the relative positions of planes and lines with the ranks of a linear system and with the scalar triple product.
- Construct with the pencil of planes the planes that contain a line and satisfy another condition.
1. The line through two points. Through two distinct points $A$ and $B$ there passes a single line, $A + t \vec{AB}$ .
  Uses: 2. Vectors and the standard basis. Proved in: Lines and planes in space.
2. Combinations of two vectors. If $\vec{u}$ and $\vec{v}$ are neither zero nor parallel, $\vec{w}$ is a combination of them if and only if $\left( \vec{u} \times \vec{v} \right) \cdot \vec{w} = 0$ , and then the coefficients are unique.
  Uses: 9. Perpendicularity, 15. The scalar triple product is a determinant. Rests on: F (Rouché-Capelli), E (Matrices and determinants). Proved in: Lines and planes in space.
3. Cartesian equation of a plane. A point lies in the plane $A + \lambda \vec{u} + \mu \vec{v}$ if and only if $\left( \vec{u} \times \vec{v} \right) \cdot \vec{AP} = 0$ ; every plane has an equation $ax + by + cz = d$ , unique up to a factor, and every such equation describes a plane.
  Uses: 12. Zero product and parallelism, 20. Combinations of two vectors. Rests on: H (Perpendiculars to two vectors). Proved in: Lines and planes in space.
4. Positions of planes. The solutions of a consistent system in $x$ , $y$ and $z$ form a point, a line or a plane according as the rank is $3$ , $2$ or $1$ ; from this follow the positions of two and of three planes and those of a line and a plane.
  Uses: 19. The line through two points, 20. Combinations of two vectors, 21. Cartesian equation of a plane, 15. The scalar triple product is a determinant. Rests on: F (Rouché-Capelli), E (Matrices and determinants), H (Perpendiculars to two vectors). Proved in: Lines and planes in space.
5. Positions of two lines. Two nonparallel lines $A + t \vec{u}$ and $B + \lambda \vec{v}$ intersect if $\vec{AB} \cdot \left( \vec{u} \times \vec{v} \right) = 0$ and are skew otherwise; that scalar triple product is zero if and only if they lie in a plane.
  Uses: 19. The line through two points, 15. The scalar triple product is a determinant, 21. Cartesian equation of a plane. Rests on: F (Rouché-Capelli), E (Matrices and determinants), H (Perpendiculars to two vectors). Proved in: Lines and planes in space.
6. Pencil of planes. The planes that contain the line common to two intersecting planes are the combinations of their equations with coefficients not both zero.
  Uses: 20. Combinations of two vectors, 22. Positions of planes. Proved in: Lines and planes in space.
Segment 4 Measurements
- Prove the formulas for the distances from a point to a plane and to a line, with the foot of the perpendicular as the only nearest point.
- Find the common perpendicular to two skew lines, prove that it is unique and compute its length with a scalar triple product.
- Define the angles between lines and planes, prove that they coincide with the geometric ones and relate the area of a plane figure to that of its projection.
- Study the sphere, its intersection with a plane and its tangent plane, and apply all of this to crystallography and to satellite positioning.
1. Distance to a plane. The distance from $(x_0, y_0, z_0)$ to the plane $ax + by + cz = d$ is $\frac{\abs{a x_0 + b y_0 + c z_0 - d}}{\sqrt{a^2 + b^2 + c^2}}$ and is attained only at the foot of the perpendicular; from it come the half-spaces and the reflection of a point.
  Uses: 21. Cartesian equation of a plane. Rests on: G (Orthogonal projection). Proved in: Distances, angles and volumes in space.
2. Distance to a line. The distance from $P$ to the line $A + t \vec{u}$ is $\frac{\norma{\vec{AP} \times \vec{u}}}{\norma{\vec{u}}}$ and is attained only at the foot of the perpendicular.
  Uses: 10. Lagrange’s identity, 19. The line through two points. Rests on: G (Orthogonal projection). Proved in: Distances, angles and volumes in space.
3. Common perpendicular. Two skew lines, $A + t \vec{u}$ and $B + s \vec{v}$ , have a single common perpendicular, and the distance between them is $\frac{\abs{\vec{AB} \cdot \left( \vec{u} \times \vec{v} \right)}}{\norma{\vec{u} \times \vec{v}}}$ , which is attained only between its feet.
  Uses: 10. Lagrange’s identity, 15. The scalar triple product is a determinant, 23. Positions of two lines. Rests on: E (Matrices and determinants), H (Perpendiculars to two vectors), G (Orthogonal projection). Proved in: Distances, angles and volumes in space.
4. Angles with planes and dihedral angles. The angle of a line with a plane, with sine $\frac{\abs{\vec{u} \cdot \vec{n}}}{\norma{\vec{u}} \norma{\vec{n}}}$ , is the one it forms with its projection, and the angle between the normals of two planes is, up to a supplement, that of the dihedral angle.
  Uses: 4. Cauchy-Schwarz and the triangle inequality, 11. Area of the parallelogram, 13. Vector triple product, 15. The scalar triple product is a determinant, 25. Distance to a plane. Rests on: D (Trigonometry), H (Perpendiculars to two vectors), G (Orthogonal projection). Proved in: Distances, angles and volumes in space.
5. Area of the projection. If a polygon of area $S$ lies in a plane that forms the dihedral angle $\alpha$ with the plane $\pi$ , with $\alpha = 0$ if they are parallel, its orthogonal projection onto $\pi$ has area $S \cos \alpha$ .
  Uses: 11. Area of the parallelogram, 15. The scalar triple product is a determinant, 25. Distance to a plane, 28. Angles with planes and dihedral angles. Rests on: B (Elementary areas and volumes), H (Perpendiculars to two vectors). Proved in: Distances, angles and volumes in space.
6. The sphere and its tangent plane. A plane at distance $\delta$ from the center of a sphere of radius $R$ cuts it in a circle of radius $\sqrt{R^2 - \delta^2}$ if $\delta < R$ and touches it only at the foot of the perpendicular if $\delta = R$ ; the tangent plane at $T$ has normal $\vec{CT}$ .
  Uses: 1. Distance between two points, 25. Distance to a plane. Rests on: C (Geometry of the plane), G (Orthogonal projection). Proved in: Distances, angles and volumes in space.

Toward calculus. The chain ends where the elementary tools end. The final review of the course places it among the seven global objectives; Calculus III will replace planes by planes tangent to curved surfaces and the determinant by the volume factor of a transformation.