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Vector Orthogonal to a Plane Determined by 3 Points
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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 a vector orthogonal to the plane defined by
A
(1, 1, –1),
B
(–2, 2, 1),
C
(1, 0, 3).
To visualize the problem, let's draw a diagram. On paper, see if you can plot the point
A
. Use an
x
-axis out of the plane of the diagram. Then check your graph by clicking on "Next".
The three components are shown in blue, yellow and green.
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Add point B. Then click "Next" to check.
The three components are shown in blue, yellow and green.
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Add point C in a similar way.
The three components are shown in blue, yellow and green.
alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag! Plot of the three points.
How do we proceed?
We can determine two vectors, AB, and AC in the plane. By taking their cross product, we can obtain a vector orthogonal to their plane.
For the vector
AB
, should the arrow point be at A or B?
B.
Add the vector
AB
to your diagram. Then click "Next" to check.
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Add the vector
AC
to your diagram. Then click "Next" to check.
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Can we use the magnitudes of these vectors and the angle between them to get the cross product?
No.
Why?
We don't know the angle between them.
How can we get the cross product?
We can determine the components of each vector and use the determinant method.
How do we get the
x
-component of
AB
?
We simply subtract the
x
-coordinates of the two points.
What is the order of the subtraction?
For the
x
-component of
AB
, we do the subtraction
Substitute values.
We get
Find the other components of this vector.
and
Determine the 3 components of
AC
.
Set up the determinant for the cross product of
AB
and
AC
in terms of the literal components
.
Substitute the values.
Set up the multiplication.
Note the sign before the
j
-component.
Simplify.
Add this vector,
, to the last diagram, with its base at point A.
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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