# Thread: Parametric equations of the plane

1. ## Parametric equations of the plane

I need help finding it. The question is below:

13. A plane has normal vector n = [3, -1, 4] and passes through the point A(1, 2, 5).

a) Determine the scalar equation of the plane.
b) Determine parametric equations of the plane.
For a I got 3x-y+4z-20=0, which is correct. However, for the parametric equation I got, x = 1+3t, y=2-t, and z=5+4t. The book says this is wrong. In fact it has two variables in it, s and t. Where do they get the second variable? How do they do part b?

Also how does one know if a line and a plane intersect? For example, how would you show that ((x+4)/2)=((y+2)/3)=99z-3)/2) intersects with plane 3x-y+2z-3=0.

2. Originally Posted by Barthayn
I need help finding it. The question is below:

13. A plane has normal vector n = [3, -1, 4] and passes through the point A(1, 2, 5).

a) Determine the scalar equation of the plane.
b) Determine parametric equations of the plane.
For a I got 3x-y+4z-20=0, which is correct. However, for the parametric equation I got, x = 1+3t, y=2-t, and z=5+4t. The book says this is wrong. In fact it has two variables in it, s and t. Where do they get the second variable? How do they do part b?
You have already done the hard part!

Now we can look at this a one equation with three unknowns so it will have to free parameters.

$3x-y+4z-20=0$ you can pick any two of them so I will let $x=s,y=t$ subbing these into the above and solving for z gives

$\displyasytle 3s-t+4z-20=0 \iff z=\frac{t-3s+20}{4}=\frac{1}{4}t-\frac{3}{4}s+5$

Now we know the solution is an ordered triple

$\displyasytle (x,y,z)=\left(s,t, \frac{1}{4}t-\frac{3}{4}s+5\right)$

Also note that parametric equations are not unique you can always check that they work by plugging back into the original plane..

3. The answers they gave were not just x=s, y=t. They were x=1+6s+t, y=2-2s+23t, and z=5-5s+5t. I understand how you got your answer, but how did thre book get theirs? I don't think my teacher will like (mark be wrong) your answer.

Also how does one know if a line and a plane intersect? For example, how would you show that ((x+4)/2)=((y+2)/3)=99z-3)/2) intersects with plane 3x-y+2z-3=0 without doing the mathematics in the first place. My teacher likes using mathematical logic statements at the beginning of the question as showing that it intersects or not.

Thanks

4. Originally Posted by Barthayn
The answers they gave were not just x=s, y=t. They were x=1+6s+t, y=2-2s+23t, and z=5-5s+5t. I understand how you got your answer, but how did thre book get theirs? I don't think my teacher will like (mark be wrong) your answer.

Also how does one know if a line and a plane intersect? For example, how would you show that ((x+4)/2)=((y+2)/3)=99z-3)/2) intersects with plane 3x-y+2z-3=0 without doing the mathematics in the first place. My teacher likes using mathematical logic statements at the beginning of the question as showing that it intersects or not.

Thanks
I just checked the answer you gave for part a) (I assumed it was correct because you said it was) and their is something wrong with it.

You gave the solution.

$3x-y+4z-20=0$ this plane is supposed to contain the point
$(1,2,5)$ but when you plug this into the plane it gives

$3(1)-(2)+4(5)-20=1 \ne 0$

So something fishy is going on!

5. Sorry bud, it was suppose to be -21 not -20. It was a typo on my part.