Subsection 19.5

Applications (19.34-19.49). The problems below, numbered 19.34-19.49, relate to a line, a circle, a plane, and a sphere. One should take the algebraic equation of the corresponding set as the definition and apply the theory of systems of linear equations

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Problem 19.34

(1)(2)

(1)

(р) Formulate in terms of ranks and prove a condition on the Cartesian coordinates $\left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right), \left(a_{3}, b_{3}\right)$ of three points of the plane, necessary and sufficient for these points not to lie on one line.

(2)

Solve the same question for four points of the plane.

Problem 19.35

Prove that through two distinct points with Cartesian coordinates $\left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right)$ there passes a unique line, and find its equation.

Problem 19.36

Show that through three points with coordinates $\left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right), \left(a_{3}, b_{3}\right)$ , not lying on one line, there passes a unique circle, and find its equation. The coordinate system is Cartesian rectangular.

Problem 19.37

(1)(2)

(1)

Three lines are given in the plane in a general Cartesian coordinate system by the equations $A_{i} x + B_{i} y + C_{i} = 0$ , $i = 1, 2, 3$ . Formulate in terms of ranks and prove a condition on the coefficients of the equations, necessary and sufficient for these lines not to pass through one point.

(2)

Solve the same question for four lines.

Problem 19.38

(1)(2)

Using the result of problem 19. 37, determine whether the given lines have a common point:

(1)

$2x + 3y + 1 = 0, 7x + 11y + 4 = 0, 3x + 4y + 1 = 0$

(2)

$x + 8y + 1 = 0, 7x - y + 1 = 0, 11x - 26y - 1 = 0$ , $8x + 7y + 2 = 0$ .

Problem 19.39

(1)(2)

(1)

Four points are given by their Cartesian coordinates $\left(a_{i}, b_{i}, c_{i}\right), i = 1, 2, 3, 4$ . Formulate in terms of ranks and prove a condition, necessary and sufficient for these points not to lie in one plane.

(2)

Solve the same question for five points.

Problem 19.40

(1)(2)

Using the result of problem 19. 39, determine whether the given points lie in one plane:

(1)

$(7, - 1, 2), (2, 3, 1), (0, 10, 0), (3, 4, 1), (6, - 2, 2)$

(2)

$(6, 1, 2), (2, 3, 1), (3, 4, 1), (6, 2, 2)$ .

Problem 19.41

Show that through four points with coordinates $\left(a_{i}, b_{i}, c_{i}\right), \quad i = 1, 2, 3, 4$ , not lying in one plane, there passes a unique sphere, and find its equation. The coordinate system is Cartesian rectangular.

Problem 19.42

(1)(2)

Three points are given by their Cartesian coordinates $\left(a_{i}, b_{i}, c_{i}\right), i = 1, 2, 3$ .

(1)

(р) Formulate in terms of ranks and prove a condition, necessary and sufficient for these points not to lie on one line.

(2)

Solve the same question for $m$ points $(m \geqslant 4)$ .

Problem 19.43

(1)(2)

Using the result of problem 19. 42, determine whether the given points lie on one line:

(1)

$(2, 3, 1), (3, 4, 2), (0, 1, - 1), (- 2, - 1, - 3), (- 6, - 5, - 7)$

(2)

$(2, 3, 1), (3, 4, 2), (0, 1, 1), (2, 1, 3), (6, 5, 4)$ .

Problem 19.44

Prove that through three points with Cartesian coordinates $\left(a_{i}, b_{i}, c_{i}\right), i = 1, 2, 3$ , not lying on one line, there passes a unique plane, and find its equation.

Problem 19.45

(1)(2)(3)

Two planes are given in a general Cartesian coordinate system by the equations $A_{i} x + B_{i} y + C_{i} z + D_{i} = 0, i = 1, 2$ . Formulate in terms of ranks and prove the conditions on the coefficients of the equations, necessary and sufficient for these planes to:

(1)

coincide

(2)

have a unique common line

(3)

be parallel, but not coincide.

Problem 19.46

Three planes are given in a general Cartesian coordinate system by the equations $A_{i} x + B_{i} y + C_{i} z + D_{i} = 0, i = 1, 2, 3$ . Formulate in terms of ranks and prove the conditions on the coefficients of the equations, necessary and sufficient for these planes to:

(1)

coincide

(2)

have a unique common point

(3)

have a unique common line

(4)

be parallel, but not all coincide

(5)

form a prism.

Problem 19.47

(1)(2)

Using the result of problem 19. 46, determine the mutual arrangement of the planes:

(1)

$3x + 2y + 5z - 1 = 0, \quad 2x + 3y + 3z + 1 = 0$ , $9x + 16y + 13z + 1 = 0$

(2)

$x - y - z + 1 = 0, \quad 5x - 21y - 17z + 1 = 0$ , $6x - 26y - 21z + 1 = 0$ .

Problem 19.48

(1)(2)(3)(4)

Four planes are given in a general Cartesian coordinate system by the equations $A_{i} x + B_{i} y + C_{i} z + D_{i} = 0, i = 1, 2$ , 3, 4. It is known that the pairs corresponding to $i = 1, 2$ and $i = 3, 4$ define straight lines. Formulate in terms of ranks and prove the conditions on the coefficients of the equations, necessary and sufficient for these lines to:

(1)

intersect

(2)

be parallel, but not coincide

(3)

coincide

(4)

be skew.

Problem 19.49

(1)(2)

Using the result of problem 19. 48, determine the mutual arrangement of the lines:

(1)

$\begin{cases} 3x + 2y + 5z - 1 = 0 \\ - x + 2y + 3z - 1 = 0 \end{cases}$ $\begin{cases} 4x - 6y + 7z + 2 = 0, \\ 5x + 3y - 8z - 3 = 0 \end{cases}$

(2)

$\begin{cases} x + 3y + 3z - 2 = 0 \\ x - 9y - 6z - 5 = 0 \end{cases}$ $\begin{cases} 2x - 14y - 9z - 9 = 0 \\ 4x + 2y + z - 7 = 0. \end{cases}$