Enrichment, Extension, and Application

Question 1

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Exercises

  1. The line \( L_3 : \begin{cases} x = t \\ y = -2t \\ z = 2t \end{cases} \) intersects the line \( L_1 : \dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z}{-6} \) at \( A \) and the line \( L_2 : \dfrac{x - 11}{2} = \dfrac{y + 8}{3} = \dfrac{z - 2}{-6} \) at \( B \). Find the vector equation of a line \( L_4 \) that intersects the lines \( L_1 \) and \( L_2 \) at \( D \) and \( C \), respectively, such that \( ABCD \) is a rhombus. (This problem has two possible solutions for \( L_4 \) — find them both.)
  2. Trapezoid \( ABCD \) has area \( 195 \), with \( AB \parallel CD \). Given \( A~(1, 0, 2), B~(4, -4, 14) \), and \( D~(9, 6, 2) \), find the coordinates of the vertex \( C \), and write the vector equation of the line \( BC \). (Hint: see if this trapezoid has a right angle.)
  3. Show that the lines with equations \begin{align*} L_1 : (x, y, z) &= (5, 3, 16) + t(1, 2, 5) \\ L_2 : (x, y, z) &= (1, 5, 10) + k(-1, 3, 2) \\ L_3 : (x, y, z) &= (-1, 1, 0) + m(3, -4, 1) \end{align*} intersect pairwise at \( 3 \) points which form a right angled triangle. Determine the area of the triangle.
  4. Given the points \( A~(1, -4, 2), B~(1, 0, 0) \), and \( C~(4, -4, 1) \), find a point \( D \) on the line \( \dfrac{x - 3}{2} = \dfrac{y - 8}{-2} = z - 1 \) such that the volume of the tetrahedron \( ABCD \) is \( 14 \). (There are two possible positions for \( D \) — find them both.)
  5. We are given the points \( A~(1, 2, 0) \) and \( B~(3, 1, 1) \). Find the point \( C \) on the line \( \begin{cases} x = 2 - k \\ y = 2 + k \\ z = 5 + 2k \end{cases} \) and the point \( D \) on the line \( \begin{cases} x = 7 - 2t \\ y = 2 - t \\ z = 4 - 3t \end{cases} \) such that the points \( A, B, C, D \) (in this order) form a parallelogram. Find the coordinates of the points \( C \) and \( D \).
  6. Parametric equations of two of the edges of a tetrahedron are given by \( AB : \begin{cases} x = 2 - t \\ y = 2 \\ z = -6 + 3t \end{cases},\ 1 \leq t \leq 2 \) and \( CD: \begin{cases} x = 1 - p \\ y = -2 + p \\ z = 4 + 2p \end{cases},\ -2 \leq p \leq 3 \). Find similar equations for the other four edges.
  7. The point \( A \) is on the half-line \( \begin{cases} x = t \\ y = -11 + 2t \\ z = 13 - 3t \end{cases}, t \geq 0 \) and point \( B \) is on the half-line \( \begin{cases} x = p \\ y = p \\ z = -1 + p \end{cases}, p \geq 0 \). Denote \( O \) as the origin of the system of axes. Find \( A \) and \( B \) such that \( OA \) is perpendicular to \( OB \) and both \( A \) and \( B \) have integral coordinates. (There are two possible solutions — find them both.)
  8. The lines \( L_1 : 3x - 4y + 7 = 0 \) and \( L_2 : 5x + 12y + 17 = 0 \) intersect at the point \( A~\left( -\frac{19}{7}, -\frac{2}{7} \right) \). Find two points \( P_1 \) and \( P_2 \) on the line \( y = 2x + 8 \) such that \( AP_1 \) and \( AP_2 \) bisect the two angles formed by the lines \( L_1 \) and \( L_2 \). (Hint: the coordinates of \( A \) are unnecessary; use the distance from the points to the two lines.)
  9. Cube ABCDHEFG as described in question Consider the cube \( ABCDHEFG \) of edge length \( 2 \). \( M \) is the midpoint of \( AB \), \( N \) is the midpoint of \( BC \), \( P \) the midpoint of \( CD \), \( R \) the midpoint of \( AE \), and \( S \) the midpoint of \( HG \). Furthermore, let \( A \) be the origin of the axes of the coordinate system, with \( AB \) lying along the \( x \)-axis and \( AD \) along the \( y \)-axis.
    1. Show that \( SN \) and \( RM \) intersect; therefore, they are situated in the same plane. Find the area of the quadrilateral \( SNMR \).
    2. Prove that the lines \( SN \) and \( MP \) are not situated in the same plane and find the distance between these two lines.
  10. Tetrahedron as described in question The edges of the regular tetrahedron \( ABCD \) all have length \( 2 \), with \( B \) at the origin of the axes, \( BC \) along the \( x \)-axis, and \( \triangle{BCD} \) in the first quadrant.
    1. Find the coordinates of the vertices of the tetrahedron.
    2. Write vector equations for the lines connecting the midpoints of opposite edges.
    3. Prove that the three segments connecting the midpoints of opposite edges of a tetrahedron are concurrent — that is, they all have a point in common.
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