Line 1 intersects both the x-axis and the y-axis. Line 2 intersects only the z-axis. Neither contains the origin. Must the two lines be parallel or skew, or can they intersect?

Find the intersection point of the following pairs of lines.

\displaystyle \begin{array}{lllll} & \phantom{.} 2x+ 5y + 15 = 0 \\ & \phantom{.} 3x -4y + 11 = 0 \end{array}

Find the intersection point of the following pairs of lines.

\displaystyle \begin{array}{lllll} & \phantom{.} \vec{r} = (-3, -6) + s(1, 1) \\ & \phantom{.} \vec{r} =(4, -8) + t(1, 2) \end{array}

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle \frac{x-3}{10} = \frac{y - 8}{-4}

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle x = 6 -18s, y = 12 + 3s\\ x = 8-6t, y = 4+ 9t

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle x =8 + 12s, y = 4-4s, z = 3- 6s \\ x = 2 - 4t, y = 2+ t, z = 6 + 2t

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle \frac{x+ 4}{3} = \frac{y - 12}{4} = \frac{z - 3}{6} \\ \frac{x}{\frac{1}{2}} = \frac{y - 10}{\frac{2}{3}} = z + 5

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \vec{r} = (-2, 0, -3) + t(5, 1, 3) \\ & \vec{r} = (5, 8, -6) + u(-1, 2, -3) \end{array}

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & x = 1 + t, y = 1+ 2t, z 1- 3t \\ & x = 3 - 2u, y = 5- 4u, z = -5 + 6u \end{array}

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \vec{r} = (2, -1, 0) + t(1, 2, -3) \\ & \vec{r} = (-1, 1, 2) +u(-2, 1, 1) \end{array}

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \vec{r} = (1 + t, 2 + t, -t) \\ & \vec{r} = (3- 2u, 4 -2u, -1 + 2u) \end{array}

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \frac{x - 3}{4} = y - 2 = z - 2 \\ & \frac{x - 2}{-3} = \frac{y + 1}{2} = \frac{z - 2}{-1} \end{array}

Consider the lines \vec{r} = (1, -1, 1)+ t( 3, 2, 1) and \vec{r} = (-2, -3, 0) + u(1, 2, 3).

• Find a vector equation for the line perpendicular to both of the given lines that passes through their point of intersection.

Consider the lines \vec{r} = (1, -1, 1)+ t( 3, 2, 1) and \vec{r} = (-2, -3, 0) + u(1, 2, 3).

• Find a vector equation for the line perpendicular to both of the given lines that passes through their point of intersection.

Show that the lines \vec{r} = (4, 7, -1) + t(4, 8, -4) and \vec{r} = (1, 5, 4)+ u(-1, 2, 3) intersect at right angles and find the point of intersection.

If they exist, find the x, y, and z intercepts of the line x = 24+ 7t, y = 4+ t, z = -20 -5t.

Find the point at which the normal through the point (3, -4) to the line 10x + 4y - 101 = 0 intersects the line.

What are the possible ways that three lines in a plane can intersect? Describe them all with diagrams.

What are the possible ways that three lines in space can intersect? Describe them all with diagrams.

Find the equation of the line through the point (-5, -4, 2) that intersects the line at \vec{r} = (7, -13, 8) + t(1, 2, -2) at 90°. Determine the point of intersection.

Find the points of intersection of the line \vec{r} = (0, 5, 3) + t(I, -3, -2) with the sphere x^2 + y^2 + z^2 = 6. Is the segment of the line between the intersection points a diameter of the sphere?