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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Find the distance from the point A(1,6,2) to the plane
P2:
Lets graph this data on paper. Draw the x-axis out of the plane of the paper, as usual. First plot point A. Then check your graph by clicking on Next.
In this diagram, note the colors used to show the three coordinates.
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 new graph. Then check your graph by clicking on Next.
Find the y-intercept in a similar way.
Add this point to your diagram. Then check your graph by clicking on Next.
Find the z-intercept in a similar way.
Add this point to your diagram. Then check your graph by clicking on Next.
Connect these three points to show the lines of intersection of the given plane with the coordinate planes.
How do we proceed?
We need another diagram. Start a 2-Dimensional diaram on paper showing the view looking parallel to the plane (the plane will show as a line). Include point A and a general point B lying in the plane. Then check your work by clicking on Next.
Add the distance D from point A to the plane. Then check your work by clicking on Next.
Add the normal to the plane, n, at point B. Then check your work by clicking on Next.
Add the vector b from point B to Point A. Then check your work by clicking on Next.
Label the angles th between b and the vertical. Then check your work by clicking on Next.
Express the distance D in terms of the magnitude of b and the angle th.
For comparison, express the Dot Product of b and n.
So we see that we can simply divide the Dot Product by
n to get
Lets use the point (0,0,4) on the plane for point
B. Set up
b in terms of the coordinates
We get
Substitute values.
How can we get the components of n?
The components are the same as the coefficients
a,b,c in the equation of the plane
Express n in terms of the values of a,b,c.
Set up the Dot Product
Simplify this.
Is this the distance we are seeking?
No.
What do we need to do further?
We need to divide by the magnitude of n.
Set up the determination of the magnitude of
n in terms of
Substitute and simplify.
Finally, determine the distance, D.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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