2. Equations and Intersections of Lines in R2 and R3
  3. Distance From a Point to a Line in \( \mathbb{R}^2 \) and \( \mathbb{R}^3 \)

Exercises


  1. For the following, find the distance between the given point \( P \) and the line \( \ell \).
    1. \( P~(1, 2) ; ~~ \ell : 3x + y - 12 = 0 \)
    2. \( P~(5, -3) ; ~~ \ell : (-8, 2) + t(2, 1) \)
    3. \( P~(4, 4) ; ~~ \ell : \begin{cases} x = -5 + t \\ y = 2 + 2t \end{cases} \)
    4. \( P~(-3, 1, 0) ; ~~ \ell : (7, -2, -3) + t(3, 2, -1) \)
    5. \( P~(0, 0, 0) ; ~~ \ell : \begin{cases} x = 12 + 3t \\ y = 7 - t \\ z = -3 + t \end{cases} \)
  2. Find the distance between each of the following pairs of parallel lines.
    1. \( \begin{aligned} \ell_1 &: (5, 2, 3) + s(3, 1, -1) \newline \ell_2 &: (-4, 2, 4) + t(3, 1, -1) \end{aligned} \)
    2. \( \begin{aligned} \ell_1 &: (0, 2, 3) + s(3, 3, 1) \newline \ell_2 &: (4, -1, 1) + t(3, 3, 1) \end{aligned} \)
    1. Find the point on the line \( \ell : \begin{cases} x = 2 + 3s \\ y = 1 - s \\ z = -4 + s \end{cases} \) that is closest to the point \( (5, -2, 8) \).
    2. Determine the distance between \( (5, -2, 8) \) and \( \ell \).
  4. The point \( A~(-5, 2, 4) \) is reflected in the line with equation \( \dfrac{x}{4} = \dfrac{y}{2} = z - 1 \). Find the coordinates of its image, \( A' \).
    1. Find the equation of \( \ell \), the line of intersection of the two planes \( \pi_1 : 2x - 3y + 4z = 3 \) and \( \pi_2 : 2x + 3y - 2z = -3 \).
    2. Determine the point on \( \ell \) that is closest to the point \( P~(2, 1, -2) \) and find the distance between them.
  6. Find the distance between each of the following pairs of skew lines.
    1. \( \begin{aligned} \ell_1 &: (4, 1, 0) + s(1, 3, 2) \newline \ell_2 &: (-5, 3, 3) + t(-1, 1, 2) \end{aligned} \)
    2. \( \ell_1 : \dfrac{x}{3} = \dfrac{y - 4}{4} = z - 8 \qquad \qquad \ell_2 : \dfrac{x + 1}{2} = \dfrac{y - 5}{5} = \dfrac{z + 2}{2} \)