Find the vector equation, parametric
equations, and symmetric equations for the line passing
through the points one, negative two, negative three
and four, negative five, negative one.
Looking at the notes below, the vector equation of a line
is given in this form where the vector R sub zero
is a positioned vector, which is a vector that points
to a point on a line and the vector V is a direction
vector, which is a vector parallel to the line.
We also have the parametric equations of a line
as well as the symmetric equations of a line.
Let's first focus on the vector equation and before we find
the vector equation let's look at this graphically.
The two given points are these two black points
and therefore the goal is to find the equation of this
red line.
If we pause this graph, notice how this blue vector
has its terminal point on the line and therefore
the blue vector is a positioned vector and notice how
the purple vector is parallel to the red line
and therefore the purple vector is a direction vector.
And remember to find the vector equation, we do need
a position vector as well as a direction vector.
So going back to our work, for the positioned vector,
vector R sub zero, let's use the vector with its initial
point at the origin and the terminal point of this first
point and therefore the vector R sub zero has an X
component of one, a Y component of negative two,
and a Z component of negative three.
Now for the direction vector, vector V, let's find
the component form of the vector that would have its
initial point at the first point and terminal point
at the second point.
And therefore in component form, the X component
is going to be four minus one.
The Y component is going to be negative five minus
negative two, and the Z component is going to be negative
one minus negative three.
Simplifying, the X component is four minus one,
which is three, the Y component is negative five minus
negative two, which is equal to negative five plus two,
or negative three.
And the Z component is equal to negative one minus
negative three, which is equal to negative one plus three,
or positive two.
This is all the information we need in order to find
the vector equation of the line passing through the two
given points.
Just keep in mind that the positioned vector can be any
vector with its terminal point on the line and the direction
vector can be any parallel vector.
So one possible vector equation would be R of T is
equal to vector R sub zero with components one,
negative two, negative three, plus T times vector V
that has components three, negative three, two.
So this is the vector equation of the line where we can
also perform multiplication here and then add
the corresponding components.
Let's also show this form of the vector equation.
So we can also express R of T with an X component
of one plus T times three, or one plus 3T.
A Y component of negative two plus T times negative three,
which gives us negative two minus 3T and a Z component
of negative three plus T times two which gives us
negative three plus 2T.
Either form of the vector equation are correct
for the line passing through the two given points.
And now that we have a vector equation for the line,
let's find the parametric equations and asymmetric
equations.
The parametric equations are given here which are found
by using the X Y and Z components of the vector equation
and therefore the parametric equations are X equals
one plus 3T, Y equals negative two minus 3T,
and Z equals negative three plus 2T.
The symmetric equations of a line are given here,
which can be found by solving each of the parametric
equations for T and then setting them equal to each other.
Looking at the X component, if we saw this for T,
we would subtract one on both sides and then divide
both sides by three, which gives us the quantity
X minus one divided by three and this is going to be
equal to solving the Y component for T, we would add two
to both sides and divide by negative three, which gives
us the quantity Y plus two divided by negative three
and this is equal to, and if we solve the Z component
for T we would add three to both sides and then divide
both sides by two which gives us the quantity Z plus three
divided by positive two and these are symmetric equations
for the line passing through the two given points.
I hope you found this helpful.