Let the lines $r \equiv \left\{ \begin{array} { l } x + y + 2 = 0 \\ y - 2 z + 1 = 0 \end{array} \right.$ and $s \equiv \left\{ \begin{array} { l } x = 2 - 2 t \\ y = 5 + 2 t \\ z = t \end{array} , t \in \mathbb { R } \right.$. a) (1.5 points) Study the relative position of the given lines and calculate the distance between them. b) ( 0.5 points) Determine an equation of the plane $\pi$ that contains the lines $r$ and $s$. c) (0.5 points) Let P and Q be the points on the lines $r$ and $s$, respectively, that are contained in the plane with equation $z = 0$. Calculate an equation of the line passing through points $P$ and $Q$.

Given the points $\mathrm { A } ( 1 , - 2,3 ) , \mathrm { B } ( 0,2 , - 1 )$ and $\mathrm { C } ( 2,1,0 )$. Find:\ a) (1.25 points) Verify that they form a triangle $T$ and find an equation of the plane containing them.\ b) ( 0.75 points) Calculate the intersection of the line passing through points A and B with the plane $z = 1$.\ c) ( 0.5 points) Determine the perimeter of triangle T.

Let the plane $\pi : z = 1$, the points $\mathrm { P } ( 1,1,1 )$ and $\mathrm { Q } ( 0,0,1 )$ and the line $r$ passing through points P and Q.\ a) ( 0.25 points) Verify that points P and Q belong to the plane $\pi$.\ b) (1 point) Find a line parallel to $r$ contained in the plane $z = 0$.\ c) (1.25 points) Find a line passing through P such that its orthogonal projection onto the plane $\pi$ is the line $r$, and it forms an angle of $\frac { \pi } { 4 }$ radians with it.

Given the plane $\pi : x + 3 y + 2 z + 14 = 0$ and the line $r \equiv \left\{ \begin{array} { l } x = 2 \\ z = 5 \end{array} \right.$, find:\ a) ( 0.5 points) Find the point on the plane $\pi$ closest to the origin of coordinates.\ b) (1 point) Calculate the orthogonal projection of the OZ axis onto the plane $\pi$.\ c) (1 point) Find the line with direction perpendicular to $r$, contained in $\pi$, and intersecting the OZ axis.

Given the points $A ( 0,0,1 ) , B ( 1,1,0 ) , C ( 1,0 , - 1 ) , D ( 1,1,2 )$, it is requested:
a) ( 0.75 points) Verify that the points $A , B , C$ and $D$ are not coplanar and find the volume of the tetrahedron they form.
b) ( 0.75 points) Find the area of the triangle formed by the points $B , C$ and $D$ and the angle $\hat { B }$ of the same.
c) (1 point) Find one of the points $E$ in the plane determined by $A , B$ and $C$ such that the quadrilateral $A B C E$ is a parallelogram. Find the area of said parallelogram.

Let the points $A ( 0,0,0 )$ and $B ( 1,1,1 )$, and the line $r \equiv ( x , y , z ) = ( \lambda , \lambda , \lambda + 1 ) , \lambda \in \mathbb { R }$. a) (1 point) Find an equation of the plane with respect to which the points $A$ and $B$ are symmetric. b) (1 point) Find an equation of the plane that contains the line $r$ and passes through the point $B$. c) (0.5 points) Find an equation of a line that is parallel to $r$ and passes through $A$.

Given the three planes $\pi _ { 1 } : - 2 x - 2 y + z = 0$; $\pi _ { 2 } : - 2 x + y - 2 z = 0$ and $\pi _ { 3 } : x - 2 y - 2 z = 0$, it is requested: a) (1 point) Determine the angle formed by the planes pairwise. Determine the intersection of the three planes. b) (1.5 points) Determine the point $P$ in space such that its orthogonal projection onto $\pi _ { 1 }$ is the point $Q _ { 1 } ( 1 / 3,4 / 3,10 / 3 )$ and its orthogonal projection onto $\pi _ { 2 }$ is the point $Q _ { 2 } ( - 1 / 3,8 / 3,5 / 3 )$. Determine the orthogonal projection $Q _ { 3 }$ of the point $P$ onto the plane $\pi _ { 3 }$.

A unit cube $ABCD-EFGH$ with edge length 1. Point $P$ is the midpoint of edge $\overline{CG}$. Points $Q$ and $R$ are on edges $\overline{BF}$ and $\overline{DH}$ respectively, and $A$, $Q$, $P$, $R$ are the four vertices of a parallelogram, as shown in the figure below.
A coordinate system is established such that the coordinates of $D$, $A$, $C$, $H$ are $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$ respectively, and $\overline{BQ} = t$. Answer the following questions.
(1) Find the coordinates of point $P$. (2 points)
(2) Find the vector $\overrightarrow{AR}$ (express in terms of $t$). (2 points)
(3) Prove that the volume of the pyramid $G-AQPR$ is a constant (independent of $t$), and find this constant value. (4 points)
(4) When $t = \frac{1}{4}$, find the distance from point $G$ to the plane containing parallelogram $AQPR$. (4 points)

In space, there is a unit cube with edge length 1. Point $O$ is one vertex, and the remaining 7 vertices are $A, B, C, D, E, F, G$. Given that $\overline { O A } = \overline { A B } = \overline { B C } = \overline { C D } = \overline { D E } = \overline { E F } = \overline { F G } = 1$ and $\overline { O G } > 1$, select the vertex farthest from point $O$.
(1) $C$
(2) $D$
(3) $E$
(4) $F$
(5) $G$

In the three-dimensional orthogonal coordinate system $x y z$, the unit vectors along the $x , y$, and $z$ directions are $\mathbf { i } , \mathbf { j }$, and $\mathbf { k }$, respectively. Using the parameter $\theta ( 0 \leq \theta \leq \pi )$, we define two curves by their vector functions $\mathbf { P } ( \theta )$ and $\mathbf { Q } ( \theta )$ :
$$\begin{aligned} & \mathbf { P } ( \theta ) = x ( \theta ) \mathbf { i } + y ( \theta ) \mathbf { j } \\ & \mathbf { Q } ( \theta ) = \mathbf { P } ( \theta ) + z ( \theta ) \mathbf { k } \end{aligned}$$
where
$$\begin{aligned} & x ( \theta ) = \frac { 3 } { 2 } \cos ( \theta ) - \frac { 1 } { 2 } \cos ( 3 \theta ) \\ & y ( \theta ) = \frac { 3 } { 2 } \sin ( \theta ) - \frac { 1 } { 2 } \sin ( 3 \theta ) \end{aligned}$$
Here, $z ( \theta )$ is a continuous function satisfying $z ( 0 ) > 0$ and $z ( \pi ) < 0$, and the curve parametrized by $\mathbf { Q } ( \theta )$ lies on the sphere of radius 2, centered at the origin $( 0,0,0 )$ of the coordinate system. The positive direction of a curve corresponds to increasing values of the parameter $\theta$. Note that the curvature is the reciprocal of the radius of curvature. Answer the following questions.
I. As $\theta$ is varied from 0 to $\pi$, calculate the arc length of the curve parametrized by $\mathbf { P } ( \theta )$.
II. Obtain $z ( \theta )$.
III. Let $\alpha$ be the angle between the tangent of the curve parametrized by $\mathbf { Q } ( \theta )$ and the unit vector $\mathbf { k }$. Calculate $\cos ( \alpha )$.
IV. Find the curvature $\kappa _ { P } ( \theta )$ of the curve parametrized by $\mathbf { P } ( \theta )$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.
V. Let $\kappa _ { Q } ( \theta )$ be the curvature of the curve parametrized by $\mathbf { Q } ( \theta )$. Express $\kappa _ { Q } ( \theta )$ in terms of $\kappa _ { P } ( \theta )$ and $\alpha$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.

4 (See the solution/explanation page)
Consider the four points $\mathrm{O}(0,\ 0,\ 0)$, $\mathrm{A}(2,\ 0,\ 0)$, $\mathrm{B}(1,\ 1,\ 1)$, $\mathrm{C}(1,\ 2,\ 3)$ in coordinate space.
(1) Find the coordinates of the point P satisfying $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OA}}$, $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OB}}$, $\overrightarrow{\mathrm{OP}} \cdot \overrightarrow{\mathrm{OC}} = 1$.
(2) Drop a perpendicular from point P to line AB, and let H be the intersection of that perpendicular with line AB. Express $\overrightarrow{\mathrm{OH}}$ in terms of $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$.
(3) Define point Q by $\overrightarrow{\mathrm{OQ}} = \dfrac{3}{4}\overrightarrow{\mathrm{OA}} + \overrightarrow{\mathrm{OP}}$, and consider the sphere $S$ centered at Q with radius $r$.
Find the range of $r$ such that $S$ has a common point with triangle OHB. Here, triangle OHB lies in the plane containing the three points O, H, B, and consists of the boundary and its interior.
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