Equations Of Lines And Planes
(requires JavaScript)
-
Find a vector equation and parametric equations for the line through the point and parallel to the vector .
, , -
Find a vector equation and parametric equations for the line through the origin and parallel to the line , , .
-
Find parametric equations and symmetric equations for the line through the points and .
-
Find a vector equation and parametric equations for the line through the point and perpendicular to the plane .,
, , and . -
Is the line through and parallel to the line through and ?Yes.
-
- Find symmetric equations for the line that passes through the point and is parallel to the line with parametric equations , , and .
- Find the points in which the required line in part (a) intersects the coordinate planes.
- .
- , , and .
-
Find parametric equations for the line segment from to .
-
Determine whether the lines and are parallel, skew, or intersecting. Find the point of intersection, if any.
: , , ,
: , , .
Parallel. -
Determine whether the lines and are parallel, skew, or intersecting. Find the point of intersection, if any.
: , , ,
: , , .
Skew. -
Determine whether the lines and are parallel, skew, or intersecting. Find the point of intersection, if any.
: ,
: .
Skew. -
Determine whether the lines and are parallel, skew, or intersecting. Find the point of intersection, if any.
: ,
: .
-
Find an equation of the plane through the point and with normal vector .
-
Find an equation of the plane through the origin and parallel to the plane ..
-
Find an equation of the plane through the origin and the points and .
-
Where does the line through and intersect the plane ?
-
Determine whether the planes and are parallel, perpendicular, or neither. If neither, find the angle between them.Neither, the angle is approx. .
-
Determine whether the planes and are parallel, perpendicular, or neither. If neither, find the angle between them.
-
Find the equation of the plane consisting of all points that are equidistant from the points and .
-
Determine whether each statement is true or false in .
- Two lines parallel to a third line are parallel.
- Two lines perpendicular to a third line are parallel.
- Two planes parallel to the third plane are parallel.
- Two planes perpendicular to the third plane are parallel.
- Two lines parallel to the same plane are parallel.
- Two lines perpendicular to the same plane are parallel.
- Two planes parallel to the same line are parallel.
- Two planes perpendicular to the same line are parallel.
- Two planes either intersect or are parallel.
- Two lines either intersect or are parallel.
- A plane and a line either intersect or are parallel.
True, false, true, false, false, true, false, true, true, false, true. -
Find parametric equations and symmetric equations for the line of intersection of the planes and .
-
Which of the following planes are parallel? Which are identical?
:
:
:
:
and are parallel, and are identical. -
Which of the following lines are parallel? Which are identical?
: , ,
:
: , ,
:
-
Find the distance from the point to the plane .
-
Find the distance between (parallel) planes and .
-
Find equations of the planes that are parallel to the plane and units away from it.
-
Show that the lines with symmetric equations and are skew, and find the distance between these lines.
-
Find the angle between the lines
and .
-
Compute all unit vectors orthogonal to the plane .
-
Find an equation for the plane containing the point and the line ..