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Topos Uranos
English

Vectors in space and the dot product

Summary

This lesson carries over to three-dimensional space the vectors that the lesson on vectors in the plane studied with two coordinates. It fixes a system of three perpendicular axes with positive orientation, states the facts of Euclidean solid geometry that are admitted and, applying the Pythagorean theorem twice, obtains the distance between two points. It defines vectors as triples of real numbers, with their sum, their product by a number, the standard basis and the uniqueness of coordinates; it defines the norm and the dot product, proves their properties, the Cauchy-Schwarz inequality through the discriminant and the triangle inequality, and with them defines the angle between two vectors, which coincides with that of elementary geometry. It then studies orthogonality, orthogonal projection, direction cosines and the writing of the dot product as the product of a row by a column, from which it follows that matrices whose columns are perpendicular unit vectors preserve lengths and angles. The applications are the work done by a force, the law of cosines in space and the angle of the methane molecule; the problems deal with the diagonals of the cube and of the parallelepiped, with the vectors perpendicular to two given ones, with the tensions in three cables and with tetrahedra whose opposite edges are perpendicular.

Learning objectives

  1. Locate points in a Cartesian coordinate system in space, recognize their octant and compute distances, justifying the formula with the Pythagorean theorem.
  2. Operate with vectors in space, express them in the standard basis and use them to prove properties of parallelograms, parallelepipeds and tetrahedra.
  3. Compute norms, dot products, angles, orthogonal projections and direction cosines, and prove the Cauchy-Schwarz and triangle inequalities in space.
  4. Write the dot product as a product of matrices and use it to decide whether a matrix preserves lengths and angles.
  5. Model with the dot product the work done by a force, the equilibrium of concurrent forces and the angles of a molecule.

The coordinate system of space

In the lesson of the course Algebra and Geometry I on vectors in the plane, each point of the plane was identified with a pair of numbers and each displacement with another pair, and the geometry of lines, angles and distances was translated into computations with those pairs. The physical world, however, has three dimensions: the position of an airplane, that of an atom in a molecule or that of a lamp hanging from the ceiling requires three numbers. If this is so, it is worth repeating the construction with triples. Most of what was proved in the plane extends without change, because the proofs worked component by component or used only the algebraic properties of the dot product; in those cases this lesson refers to the one on the plane. What demands care is the geometry: in space there is no plane figure that contains everything, and the angle between two vectors, for instance, must be defined before it is computed.

The facts of Euclidean solid geometry that are admitted

Analytic geometry of space translates the geometry of solids, which Euclid set out in books XI to XIII of the Elements. Its axiomatic construction lies beyond this course, as happened with that of the plane in the lesson on the equation of a line, where the facts of plane geometry that are admitted were gathered in a list. In the same way, the facts about space that this unit uses are admitted without proof and gathered in the list below; whenever a lesson of the unit resorts to one of them, it refers to this list.

  1. Planes. Through three noncollinear points there passes exactly one plane, and if two points of a line lie in a plane, the whole line lies in it. In each plane of space the facts of plane geometry admitted in the lesson on the equation of a line hold; notably, the Pythagorean theorem and its converse, and the freedom to choose a system of axes in it.
  2. Perpendicular lines and planes. Through each point there passes exactly one plane perpendicular to a given line, and that plane meets the line at exactly one point. A line perpendicular to a plane is perpendicular to all the lines of the plane that pass through the point where it meets the plane.
  3. Axes. Three lines that pass through a point OO and are pairwise perpendicular, with a positive sense on each one and the same scale of length, form a system of axes. Distances, angles and figures do not depend on that choice.
  4. Coordinates. Once a system of axes is fixed, the planes perpendicular to the three axes drawn through a point PP meet them at the points with abscissas xx, yy and zz, and conversely, for each triple of numbers those three planes meet at exactly one point. The points whose third coordinate equals cc form a plane perpendicular to the zz-axis, in which the first two coordinates are Cartesian coordinates of the plane with the same scale; the points with fixed first coordinates form a line parallel to the zz-axis, perpendicular to that plane, on which the third coordinate measures position with the same scale. The same holds with the axes permuted.

The fourth fact is a consequence of the first three, which solid geometry proves with the properties of parallel lines and of parallel planes; it is stated separately because it is the form in which it will be used.

A Cartesian coordinate system in space is a system of axes, called the xx-axis, the yy-axis and the zz-axis, which meet at the origin OO. Each point PP is identified with the triple of its coordinates, P=(x,y,z)P = (x, y, z), which are called the abscissa, the ordinate and the applicate. The three coordinate planes are the xyxy-plane, with equation z=0z = 0; the yzyz-plane, with equation x=0x = 0; and the xzxz-plane, with equation y=0y = 0.

The coordinate planes divide space into eight regions, called octants, according to the sign of each coordinate; the first octant is that of the points with all three coordinates positive, and the others are named by the triple of their signs, for example the octant (+,−,+)(+, -, +). In the plane, two perpendicular axes can be arranged in two essentially different ways: the second axis is obtained by rotating the first a quarter turn counterclockwise, or clockwise. In space the same happens with the three axes, and the distinction has a name.

A coordinate system in space has positive orientation when, viewed from the positive side of the zz-axis, the quarter turn that takes the positive xx-axis to the positive yy-axis is seen counterclockwise. Equivalently, the right-hand rule holds: if the fingers of the right hand curl from the positive xx-axis toward the positive yy-axis, the thumb points along the positive zz-axis.

If the sense of one of the axes is reversed, the orientation changes; if two are reversed, it is preserved. Permuting the axes cyclically, from xx, yy, zz to yy, zz, xx, does not change the orientation either, so that in a positive system, viewed from the positive xx-axis, the rotation from yy to zz is also counterclockwise. From here on all systems have positive orientation. The figures draw them in cavalier projection: the yy-axis to the right, the zz-axis upward and the xx-axis toward the viewer, drawn diagonally and with lengths halved. Orientation plays no role in this lesson; it will be essential in the following one, on the cross product and the scalar triple product.

ExampleLocating a point in space

Locate the point P=(3,4,2)P = (3, 4, 2) by moving along the axes, state its octant and locate the point P1=(3,4,−2)P_1 = (3, 4, -2).

Proof

  1. O=(0,0,0)O = \left( 0, 0, 0 \right)

    We draw the three axes with positive orientation: seen from above, the quarter turn from the positive xx-axis to the positive yy-axis is counterclockwise.

  2. A=(3,0,0)A = \left( \resaltar{3}, 0, 0 \right)

    We move a distance 33 along the xx-axis.

  3. B=(3,4,0)B = \left( 3, \resaltar{4}, 0 \right)

    From AA, we move a distance 44 in the direction of the yy-axis, without leaving the xyxy-plane.

  4. P=(3,4,2)P = \left( 3, 4, \resaltar{2} \right)

    From BB, we rise a distance 22 in the direction of the zz-axis. The three coordinates are positive: PP lies in the first octant, and it is the vertex opposite OO of the box with edges 33, 44 and 22.

  5. P1=(3,4,−2)P_1 = \left( 3, 4, \resaltar{-2} \right)

    If from BB one goes down a distance 22, one reaches P1P_1, in the octant (+,+,−)(+, +, -). The xyxy-plane cuts the line segment PP1P P_1 perpendicularly at its midpoint, BB: P1P_1 is the reflection of PP in that plane.

The distance between two points

In the plane, the distance between two points was obtained by applying the Pythagorean theorem once to a triangle whose legs are parallel to the axes. In space it must be applied twice: first in a horizontal plane and then in a vertical plane.

TheoremDistance between two points

The distance between the points P=(x1,y1,z1)P = (x_1, y_1, z_1) and Q=(x2,y2,z2)Q = (x_2, y_2, z_2) is given by the formula below.

d⁡(P,Q)=(x2−x1)2+(y2−y1)2+(z2−z1)2\operatorname{d}(P, Q) = \sqrt{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2 + \left( z_2 - z_1 \right)^2}

An auxiliary point RR is introduced, at the height of PP and on the vertical through QQ. The figure uses P=(1,2,1)P = (1, 2, 1) and Q=(3,5,7)Q = (3, 5, 7).

Proof

  1. R=(x2,y2,z1)R = \left( x_2, y_2, \resaltar{z_1} \right)

    Let RR be the point with the first two coordinates of QQ and the applicate of PP. The points PP and RR lie in the plane of applicate z1z_1, and RR and QQ on the line parallel to the zz-axis formed by the points with first coordinates x2x_2 and y2y_2.

  2. d⁡(P,R)2=(x2−x1)2+(y2−y1)2{\operatorname{d}(P, R)}^2 = \dato{pr}{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2}

    By the fact on coordinates, in the plane of applicate z1z_1 the first two coordinates are Cartesian, and the distance formula in the plane, from the lesson on the equation of a line, gives the length of PRPR. In the figure, 4+9=134 + 9 = 13.

  3. d⁡(R,Q)=∣z2−z1∣\operatorname{d}(R, Q) = \dato{rq}{\abs{z_2 - z_1}}

    On the vertical line that contains RR and QQ, the applicate measures position: the length of RQRQ is the absolute value of the difference of the applicates. In the figure, 7−1=67 - 1 = 6.

  4. d⁡(P,Q)2=d⁡(P,R)2+d⁡(R,Q)2{\operatorname{d}(P, Q)}^2 = \resaltar{{\operatorname{d}(P, R)}^2} + \resaltar{{\operatorname{d}(R, Q)}^2}

    The line RQRQ is perpendicular to the plane of applicate z1z_1 and, by the second admitted fact, to the line PRPR, which lies in that plane and passes through RR. The triangle PRQPRQ has a right angle at RR and, by the first fact, the Pythagorean theorem holds in the plane that contains it.

  5. d⁡(P,Q)=(x2−x1)2+(y2−y1)2+(z2−z1)2,13+36=7\operatorname{d}(P, Q) = \sqrt{\resaltar{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2} + \resaltar{\left( z_2 - z_1 \right)^2}}, \qquad \sqrt{13 + 36} = 7

    We substitute and take the square root. If P=RP = R or R=QR = Q, there is no triangle, but the formula reduces to that of only one of the two distances and remains true. In the figure, the distance is 77.

As a special case, the distance from P=(x,y,z)P = (x, y, z) to the origin is x2+y2+z2\sqrt{x^2 + y^2 + z^2}, the length of the diagonal of the box with edges ∣x∣\abs{x}, ∣y∣\abs{y} and ∣z∣\abs{z}. For example, the diagonal of a cube with edge aa measures a2+a2+a2=a3\sqrt{a^2 + a^2 + a^2} = a \sqrt{3}.