Area of a Parallelogram
General Contents
Detailed Contents
Index
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The following points define a parallelogram:
A
(6, – 3, – 2),
B
(3, – 3,2),
C
( – 4,8,4), and
D
( – 1,8,0) Find its area.
How do we proceed?
We calculate the Vector Cross Product of vectors representing two sides of the parallelogram.
Must the sides be intersecting?
Yes.
Will that result be a scalar or a vector?
It will be a vector,
V
.
Is area a scalar or a vector?
It is a scalar.
How can we get the area from
V
?
The area is the magnitude of
V
.
How do we get the magnitude of
V
?
We take the square root of the sum of the squares of its components.
In order to understand the problem better, let’s graph these points 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. Point
A
is not on the
x
-axis.
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Plot point
B
. Then check your graph by clicking on “Next”.
Point
b
is not on the I>y-axis.
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Plot point
C
. 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!
Plot point
D
. 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 points to show the parallelogram. 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!
Let’s use vectors
AB
and
AD
for our cross product.
Express the
x
-component,
AB
x
, in terms of the symbols for the coordinates of the points
A
and
B:
A
x
, A
y
, A
z
, B
x
, B
y
, and
B
z
.
Evaluate this using the data from the problem.
We get
In a similar way, evaluate
AB
y
.
In a similar way, evaluate
AB
z
.
Use this information to write
AB
in terms
of the unit vectors
i
,
j
, and
k
.
We get
AB
= – 3
i
+ 0
j
+ 4
k
In a similar way, determine the three components of
AD
.
Use this information to write
AD
in terms of the unit vectors
i
,
j
, and
k
.
We get
AD
= – 7
i
+ 11
j
+ 2
k
Now we are ready to find the cross product of these two vectors. Set up
in determinant form.
Evaluate the determinant.
We get
Now set up the determination of the magnitude of
V
in terms of its components.
Substitute values and simplify.
Summarize what we have done.
The area of the parallelogram defined by the given four points is
.
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