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)
.
Consider the lines \vec{r} = (1, -1, 1)+ t( 3, 2, 1)
and \vec{r} = (-2, -3, 0) + u(1, 2, 3)
.
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?
Introduction to Vector form of a line
Finding distance using vector equation of a line
Parametric equation of a line
Finding Parametric equation of a line converting it to Catesian RectangularForm
Example of Vector Form of a Line
ex6 Finding Point of Intersection in 2D
ex7 Finding Point of Intersection in 3D
ex8 Finding intersection of two lines in 3D with no solution
Symmetric form of a line
Shortest Distance formula
Shortest distance from a point to 3D line
Shortest Distance to line from a Point in 3D
This video has an error at the end with distance calculation - forgot to square root for lengths