Equations of Lines and Planes Level 3.
In the previous video we derived the vector equation, parametric equations, and symmetric
equations of a line in space.
In this video we will go over three basic examples that make use of these 3 equations.
Alright let’s jump straight into the first example.
Determine whether each point lies on the line.
In this problem we are given parametric equations of a line in space along with the coordinates
of two points.
We are asked to determine if these points lie on the line.
Recall that a point will lie on a curve if the points’ three coordinates satisfy the
parametric equations for the same value of the parameter t.
We can determine this in a couple of ways we can substitute the coordinates of the point
into the parametric equations and solve for the parameter t for all three parametric equations.
If all three parametric equations yield the same value for the parameter, then the point
lies on the line.
On the other hand if at least one equation has a distinct parameter then we can conclude
that the point does not lie on the line.
So let’s go ahead and substitute the coordinates of the first point into the parametric equations
and solve for the parameter t, doing that we obtain the following.
Notice that all three parametric equations yielded the same parameter t this means that
the first point lies on the line.
If we had the symmetric equations of the line we would solve it in a similar manner in this
case we would substitute the points’ coordinates and check to see if all three equations yield
the same value.
We can also show that a point lies on the line by determining the parameter t for one
of the points coordinates and use it to check if the parameter satisfies the two other parametric
equations.
Let’s illustrate this with the second point.
Let’s first determine the parameter t for the first parametric equation so we substitute
the x-coordinate into the first parametric equation and solve for the parameter t.
Now that we have this parameter let’s substitute it into the other two parametric equations
and check to see if we obtain the same y and z coordinate of the point in question.
Notice that the second parametric equation is not satisfied and at this point we can
stop here since we only need one parametric equation to be false to conclude that this
point is not on the line.
As a check notice that the third parametric equation is also not satisfied.
This means that the coordinates of this point occur at different values of the parameter
t unlike the first point where all coordinates exist at a common parameter t.
So the first point lies on the line and the second point does not and this is our final
answer.
Let’s take a look at the next example.
Find a vector equation, parametric equations and symmetric equations for line L that passes
through point P and is parallel to vector V.
In this problem we are asked to find the equation of a line in space using all three forms.
Let’s first determine the vector equation of the line.
Recall that the vector equation of a line is written by finding a position vector that
terminates at a point on the line (in this case the initial point) plus a parameter multiplied
by a vector parallel to the line.
Notice that we are provided with a point and it is known that line L passes through this
point; we will use this point as the terminal point of a position vector r zero.
We are also given a vector parallel to the line.
So we go ahead and substitute these vectors into the vector equation of a line in space.
Now it is just a matter of multiplying the parameter t with the components of vector
V. Then we go ahead and add the vectors on the right side of the equation component wise,
doing that we obtain the following vector equation.
Note that we can also rewrite this vector equation by using the standard unit vectors
i, j and k.
Next let’s find the parametric equations for line L. To find a set of parametric equations
we will break apart the components of our vector equation into individual equations,
one for the x-component, y-component and z-component.
Notice that the parametric equation makes use of the coordinates of the point where
the line passes through and also uses the direction numbers of vector V.
Also notice that the parametric equations are essentially the components of the vector
equation of the line.
Lastly, we can find the symmetric equations of the line by solving for the parameter t
for each parametric equation and equating all three expressions to one another, doing
that we obtain the following symmetric equations.
Notice that the symmetric equations also use the coordinates of the point where the line
passes through as well as the direction numbers of vector V. It is important to note that
the vector, parametric and symmetric equations can be obtained by simply rewriting one form
into any of the two other forms, once you know one form the other two are also known,
it is just a matter of rewriting them.
Alright let’s take a look at the final example.
Find a vector equation, parametric equations and symmetric equations for line L that passes
through point A and point B. At what point does this line intersect the xy-plane?
In this problem we are asked to find a vector, parametric and symmetric equations of a line
that passes through points A and B.
We are also asked to determine the point in which this line intersects the xy plane.
Let’s first determine the equations of the line.
Notice that we are given two points A and B but we are not provided with a vector that
is parallel to the line.
We are going to have to generate a parallel vector by using these two points.
This is very similar to the way we found the slope of a line in R squared when we were
given two points, recall that we had to find the change in y and the change in x.
For this problem we can find a vector V that is parallel to the line by finding a displacement
vector, we will use point A and point B to find this parallel vector.
So we use Point B as the terminal point and use point A as the initial point of the displacement
vector V.
We then subtract the coordinates of point A from the coordinates of point B as follows.
Doing that we obtain the following.
This is one of many possible vectors that are parallel to the line, we just happen to
use this particular vector because we can determine it quickly with the given points.
There is nothing stopping you from using a scalar multiple of this vector instead of
the one we found and in fact you might want to make the components of direction vectors
nicer to work with and this can be done by scaling them.
Alright now that we have a vector parallel to the line we can now write the vector equation
of the line.
We need to choose an initial point for our vector equation, since it really does not
matter which point we choose we are going to choose point A for our vector equation
(you are free to use point B as well).
Substituting the components of the displacement vectors’ terminal point A and the components
of the parallel vector V, we obtain the following.
Now it is just a matter of distributing the parameter t, and adding the resulting vectors,
doing that we have the following vector equation of the line.
We can also rewrite this vector equation by using unit vectors.
Next let’s find the parametric equations of the line.
We can obtain the parametric equations by isolating each of the components of this vector
equation, yielding three separate equations, one for each component as follows.
Lastly, we can generate the symmetric equations by solving for the parameter t for each parametric
equation and equating all three expressions to one another, doing that we obtain the following.
Next let’s figure out the coordinates where this line intersects the xy-plane.
We can determine the point of intersection by remembering that any point located on the
xy-plane will have a z-coordinate equal to zero.
So we can approach this in two ways, we can use the parametric equations and set z equal
to zero on the third equation and solve for the parameter t, this will provide us with
the value of t where the z-coordinate of the line is equal to zero, we then use this value
and substitute it into the other two parametric equations and solve for the x and y-coordinates.
We can also use the symmetric equations and set the variable z equal to 0.
This will generate two separate equations that can be used to solve for the x-coordinate
and y-coordinate of the point.
Using this method we obtain the following two equations, remember that we can cross
multiply proportions to rewrite them into an equivalent equation.
Solving for x and y we obtain the following.
In the end line L intersects the xy-plane at the following point, and this is our final
answer.
Alright In our next video we will take a look at slightly more challenging examples involving
the equations of lines in space.