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LatineSummary
This lesson opens the fifth and last unit of the course, devoted to the analytic geometry of space. It presents the four lessons that make it up and the thread that unites them: the vectors of the plane, studied in Course I, become triples of numbers with a dot product that measures lengths and angles; the determinant of order three from the preceding unit yields a new product, the cross product, and with it the scalar triple product, which measures areas and volumes; lines and planes are described by linear equations, and their relative positions are decided with the rank and the Rouché-Capelli theorem (known in the Spanish-speaking world as the Rouché-Frobenius theorem); and all of this comes together to measure distances and angles between points, lines and planes. It traces the historical background of these ideas, from the coordinates of Descartes and Fermat, which in the seventeenth century barely left the plane, to the vector calculus of Gibbs and Heaviside and crystallography; it formulates precisely, by means of two tunnels of a subway network, the problem that gave rise to them, that of deciding whether two lines in space intersect and of measuring how far apart they are when they do not; and it points out where they are used again in this course, in other courses on the site and in science and technology.
Learning objectives
- Describe the sequence of lessons of the unit and explain which tool each one contributes and which ones it takes from the unit on matrices and from the plane geometry of Course I.
- Place in their historical context the coordinates of space, the dot product, the cross product and vector calculus, naming the problem that each idea came to solve.
- Pose as a linear system the question of whether two lines in space intersect, decide it with the rank of the augmented matrix and compute the shortest distance between two skew lines.
- Identify uses of the analytic geometry of space in physics, computer graphics, robotics, geodesy, satellite positioning and chemistry.
Entry assessment
Before working through the unit, it is advisable to measure the starting point. The first eight questions cover, two by two, the four lessons of the unit, and the ninth is a modeling problem, which asks for a situation in context to be set up with the distance from a point to a line; the same skills are measured again at the end, in the synthesis lesson. The last three check what the unit takes as known from the unit on matrices and from Course I. There is no grade: it serves to show what has been mastered and where to slow down.
Before starting the unit, measure your starting point. It is a short self-assessment: it is not graded and its result is stored only in your browser.
Question 1Compute the distance between the points and .
Correct answer:
The differences of the coordinates are , and , and the distance is . Whoever obtains has added the differences without squaring them and without taking the square root.Question 2What angle do the vectors and form?
It comes from dividing the dot product by only one of the norms, which gives . The cosine of the angle is the dot product divided by the product of the two norms.It is the angle whose sine equals . The quotient of the dot product by the product of the norms is the cosine of the angle, not its sine.The dot product is and each vector has norm ; the cosine of the angle is , which corresponds to .Question 3Compute .
The minus sign of the second term of the expansion of the formal determinant along the first row is missing: the second component is . The vector is not perpendicular to .It is the product in the opposite order, . The cross product is anticommutative: changing the order of the factors changes the sign.It is the componentwise product, which is not perpendicular to the factors and has no geometric meaning. Each component of the cross product is a determinant of order two.The components are , and ; the result is perpendicular to both factors, as can be checked with the dot product.Question 4Compute the volume of the tetrahedron with vertices , , and .
Correct answer:
The scalar triple product of , and is the determinant with rows , and , which equals ; it is the volume of the parallelepiped, and that of the tetrahedron is one sixth of it, . Whoever obtains has computed the parallelepiped, and whoever obtains has taken half, as in the area of the triangle.Question 5What is the equation of the plane through , and ?
The intercepts with the axes have been taken as coefficients. The equation of a plane that cuts the axes at , and is ; here, , which is equivalent to . The point does not satisfy .The normal vector is correct, but the constant term has not been computed: this equation describes the parallel plane through the origin, and none of the three points satisfies it.It is the line through and . A plane needs two nonparallel direction vectors, or a normal vector.A normal vector is , proportional to ; the constant term comes from substituting : . The three points satisfy the equation.Review: Lines and planes in space
Question 6What is the relative position of the lines and ?
That is what happens in the plane with two nonparallel lines. In space, equating the points gives three equations in two unknowns, and the third, , is impossible.In the plane, two lines with no common points are parallel; in space they are not necessarily, because they can be skew. The direction vectors and are not proportional.Their direction vectors are not parallel, and the system , , has no solution; two nonparallel lines with no common points are skew. The scalar triple product of , and equals , which is nonzero, and this confirms that they do not lie in one plane.Review: Lines and planes in space
Question 7Compute the distance from the point to the plane .
Correct answer:
By the formula for the distance from a point to a plane, . Whoever obtains has forgotten to divide by the norm of the normal vector, and whoever obtains has added the constant term instead of subtracting it.Question 8What angle does the line with direction vector form with the plane ?
It is the angle of the line with the normal vector, not with the plane. The angle with the plane is the one the line forms with its projection onto it, and it is the complement of the angle with the normal.It comes from dividing the dot product by only one of the norms, which gives a cosine of . One must divide by the product of the two norms.The line would be parallel to the plane if its direction vector were perpendicular to the normal vector; but the dot product of and equals , not .The direction vector and the normal vector form an angle with cosine , that is, an angle of ; the angle of the line with the plane is its complement, .Question 9A drone takes off from the point of a park and climbs in a straight line in the direction of the vector , with distances in meters. At the point on the ground a person is sitting. What is the minimum distance, in meters, at which the drone passes that person?
Correct answer:
The trajectory is the line , with of norm , and the minimum distance is the distance from to that line: . It is attained at the foot of the perpendicular, the point , because is perpendicular to . Whoever obtains has computed the projection , which measures how far the drone travels up to that foot, and not the distance; whoever obtains has measured the distance to the takeoff point.Question 10From earlier unitsCompute the determinant of the matrix with rows , and .
Correct answer:
Expanding along the first row, . Whoever obtains has forgotten the alternating sign of the second term of the expansion.Question 11From earlier unitsA system of three linear equations in two unknowns has a coefficient matrix of rank and an augmented matrix of rank . How many solutions does it have?
That the rank of the coefficient matrix equals the number of unknowns guarantees uniqueness only if the system is consistent, and consistency requires the augmented matrix to have the same rank.There are infinitely many solutions when the system is consistent and the common rank is less than the number of unknowns. Here the system is not consistent, because the two ranks differ.By the Rouché-Capelli theorem, a system has a solution if and only if the two matrices have the same rank; here the ranks are different, and the system is inconsistent.Question 12From earlier unitsFor what value of are the vectors and of the plane perpendicular?
Correct answer:
Two nonzero vectors are perpendicular if and only if their dot product is zero: , whence . Whoever obtains has changed a sign; whoever equates the slopes is looking for parallel vectors, not perpendicular ones.Review: Vectors in the plane and parametric equations of a line
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