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Line through a Point parallel to a Vector:
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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 vector, parametric, and symmetric equations for the line L through P(2,-3,2) and parallel to
.
Let's plot this on paper, using the usual 3-D axes with the x-axis out of the plane of the diagram. To begin, plot the point P. Then check your work by clicking "Next".
Note the use of color for the coordinates of the point..
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Add the vector
a
to your diagram. Show it with its tail at the origin. Then check your work by clicking "Next".
Here the colors represent the coordinates of the point.
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Add a line parallel to
a
that goes through P.
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How do we get a vector equation for the line?
One way is to construct
1) a vector,
, from the origin to point P, and
2) a vector along the line.
Then we add the two vectors.
How do we get a vector in the direction of the line?
We can use a multiple of the vector
a
, since that is in the direction of the line L. We can call this multiple t
a
, where t is a parameter.
Show
in your diagram. Then click "Next" to check.
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Add t
a
to your diagram. Then click "Next" to check.
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Now show the vector sum
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Summarize what
r
represents.
r
describes any point on the line. The various locations of points on the line L are associated with corresponding values of the parameter t, which need not be an integer.
Relate the general vector sum to the data given in the problem.
We get the vector equation for the line L:
Collect the coefficients of
i, j, k
on the right-hand side.
This is the
vector equation
of the line.
Extract from this equation the
parametric equation
for
x
.
Equating the coefficients of
i
on each side, we get
Do the same for
y
.
Do the same for
z
How do we find the symmetric equations for the line?
We solve the parametric equations for t.
Do that.
We get
symmetric equations
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