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Hi everyone. Welcome back to integralcalc.com. Today we're going to talk about how we're
going to find the symmetric equations of a line. In our problem, we're given a point
P and a vector v and our line is represented by the point and the vector. So it's passing
through this point P (2, 3, -4) and the vector represents this direction numbers here.
So we're going to find the symmetric equations of the line that's represented by these two
things here. This is a very easy concept if we have the right information and in this
case we do. The formula that we’re going to be using is the following.
Our problem gives us the point P and the vector v and the point P (2, 3, -4) represents x
sub 0, y sub 0 and z sub 0, respectively. Our vector v inside here, we have our direction
numbers, a, b and c. a is equal to 1, b is equal to negative 1 and c is equal to negative
2. That’s always going to be the case. You’re
always going to plug in the three direction numbers inside of your vector and the three
points inside of your point P for x sub 0, y sub 0, and z sub 0.
That's going to be the solution to our problem. We plug in each of those points and then all
we do from here is simplify to get our final answer; and maybe it's simplified enough already
but in our case, we have simplification to do.
Obviously the 1 in the denominator is going to go away. That’s redundant and we don't
need it. The negative one on the denominator of the
second term is going to flip the sign on both the y term and the negative 3 terms so instead
of y minus 3, we're going to end up with negative y plus 3.
For our symmetric equation here in z, we're going to bring this negative on the denominator
out in front of the fraction and we're going to combine this double negative into a positive
so we're going to end up with negative z plus 4 divided by 2.
And that's it. That’s all we have to do. It's just basically plugging in and simplifying,
as long as you're given enough information on the original problem.
I hope that helped you guys and I'll see you in the next problem. Bye.