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Three Vectors Coplanar
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Determine whether the vectors specified by the following four points are coplanar:
A
(6, 0, 2),
B
(2, 0, 4) ,
C
(6, 6, 1) ,
D
(2, 6, 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.
Add point B. Then click "Next" to check.
The three components are shown in blue, yellow and green.
Add point C in a similar way.
The three components are shown in blue, yellow and green.
Add point D.
The three components are shown in blue, yellow and green.
How do we proceed?
Since three points, such as
determine a plane we can determine two vectors,
, in that plane. By taking their cross product, we can obtain a vector perpendicular to their plane. Then we can determine the dot product of that vector and the vector
.
How will that help?
That dot product is the volume of the parallelepiped determined by the three vectors.
What should we look for?
If the dot product is zero, the volume is zero.
And then?
We can conclude that the three vectors lie in the same 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.
Add the vectors
AC
and
AD
to your diagram. Then click "Next" to check.
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.
We get
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.
To get an idea of the direction of this vector, scale it down by a factor of 4 to
so that it will fit better on the diagram. Add this vector to the last diagram, with its base at point A.
Are we done?
No.
What else do we need to do?
We need to determine dot product of this vector with
AD
.
Determine the 3 components of
AD
.
Now set up the dot product.
We need
Substitute the vectors.
Set up the products of the components.
Simplify.
We get
What is our conclusion?
The volume is zero and the three vectors are coplanar.
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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General Contents
Detailed Contents
Index