Point of Intersection of a Line and a Plane
General Contents
Detailed Contents
Index
If you find this page helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Find the point of intersection, A, of the line, L:
with the plane P:
Lets graph this data on paper. To plot the plane, we can use its intercepts on the axes. How can we find its x-intercept?
Since
for all points on the x-axis, we can substitute those zero values into the equation of the plane and solve for
.
Do that.
Show this point on a graph. Draw the x-axis out of the plane of the paper, as usual. Then check your graph by clicking on Next.
alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Find the y-intercept in a similar way.
Add this point to your diagram. Then check your graph by clicking on Next.
alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Find the z-intercept in a similar way.
Add this point to your diagram. Then check your graph by clicking on Next.
alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Connect these three points to show the lines of intersection of the given plane with the coordinate planes.
alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Finally, we need to draw the given line. From its equations, we see that it goes through (5,7,-2) and has direction numbers 1, 3, -2. Add the line to your diagram, and then click Next to check.
Note that the line intersects the plane (triangle) at A. Eventually,we can show part of it as hidden by the triangle.
alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Now lets find the coordinates of point
A
,
, where the line intersects the plane.How are these coordinates related to the equations of the line ?
Since the point A is the intersection of the line and the plane, its coordinates must satisfy the equation of the line.
How does this help?
We can express the point on the line in terms of a single unknown,
.
How are
x
A
, y
A
, z
A
related to the equation of the equation of the plane,
?
Since point A is the intersection of the line and plane, the coordinates must satisfy the equation of the plane.
Consequently, we have four unknowns,
, and four equations relating them.
Set up the three equations from the line relating them.
Substitute these results into the equation for the plane.
Solve for
.
We get
Use this to determine the coordinates of point
A
.
We get
State the coordinates of point
A
.
(3,1,2)
Redraw the diagram showing the coordinates of point
A
and the line disappearing behind the triangle at point
A
.
alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
General Contents
Detailed Contents
Index