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# Finding an othogonal vector (Elimiante x and y) Watch

1. I'm stuck on this format of question:
For what values of x and y is vector v = [6, 2, x]T orthogonal to the plane containing points
A(1, 8, 2), B(1, −1, −4) and C(5, y, 15)? Record x then y on the same line.

I'm not sure how to solve for x and y. With out them there I'd form two vectors AB and AC then use the cross product to get an equation for the plane and compare the co-efficients with the vector but I honestly have no idea how to get rid of the x and y values... would I set them to 0 and solve, or do I need simultaneous equations?
2. (Original post by matte30)
I'm stuck on this format of question:
For what values of x and y is vector v = [6, 2, x]T orthogonal to the plane containing points
A(1, 8, 2), B(1, −1, −4) and C(5, y, 15)? Record x then y on the same line.

I'm not sure how to solve for x and y. With out them there I'd form two vectors AB and AC then use the cross product to get an equation for the plane and compare the co-efficients with the vector but I honestly have no idea how to get rid of the x and y values... would I set them to 0 and solve, or do I need simultaneous equations?
Proceed as normal(!). What do you get? It does work out.
3. (Original post by ghostwalker)
Proceed as normal(!). What do you get? It does work out.

(I think that's the best I can do for formatting; I hope it helps)
4. (Original post by ghostwalker)
Proceed as normal(!).
PRSOM
matte30, it would be really helpful if you could LaTeX your working; you can literally just write [ latex] [/ latex] around it (without the spaces), and use \times for a cross if you want it.
5. (Original post by matte30)

(Matrix removed because I'm not sure how to format it; but am working on it now)
Agree with your k component, but not the i, or j. j should be the negative of what you have.

AC = (4,y-8,15-2)

Then just compare the 2nd coordinate of your result with that of your vector v, to get the scaling factor, and ....

Edit: Corrected.
6. (Original post by ghostwalker)
Agree with your k component, but not the i, or j. j should be the negative of what you have.

AC = (4,y-8,15-2)

Then just compare the 1st coordinate of your result with that of your vector v, to get the scaling factor, and ....

Edit: Corrected.
AC = (4, y-8, 13)

I found why the j should be negative; thank you.
To get the first co-ordinate do I set y to 0?
7. (Original post by matte30)
AC = (4, y-8, 13)

I found why the j should be negative; thank you.
To get the first co-ordinate do I set y to 0?
Don't agree with the "-192i", I make it "-165i", otherwise we are in agreement.

No, you don't set y to 0.

This (corrected) vector is orthogonal to the plane, as is v.

So, one must be a scalar multiple of the other.

You want to compare the coefficients of the coordinates to work out the scaling factor. You need to check j - not i as I previously said (sorry about that) - to get the scaling factor.
8. (Original post by ghostwalker)
Don't agree with the "-192i", I make it "-156i", otherwise we are in agreement.

No, you don't set y to 0.

This (corrected) vector is orthogonal to the plane, as is v.

So, one must be a scalar multiple of the other.

You want to compare the coefficients of the coordinates to work out the scaling factor. You need to check j - not i as I previously said (sorry about that) - to get the scaling factor.
I cannot get -156i; the closet I have is -165i.
The corrected k column finishes with +13.

-9(13)i = -117i
6(y-8)i = 6yi -48i
-117 - 48 = 165

----------------------------------------------

For the actual question I've been asked I've ended up comparing my j value with a 0 in the vector... once again, I'm not sure what to do now since there is no scale factor...
9. (Original post by matte30)
I cannot get -156i; the closet I have is -165i.
The corrected k column finishes with +13.

-9(13)i = -117i
6(y-8)i = 6yi -48i
-117 - 48 = 165
Yes, you're right -165i. Clearly, I can't multiply.

So, (-165+6y)i-24j+36k

I presume you can now finish off.
10. (Original post by ghostwalker)
Yes, you're right -165i. Clearly, I can't multiply.

So, (-165+6y)i-24j+36k

I presume you can now finish off.
For this question; yes. But what if the unit you're comparing with a 0? example v is changed from [6, 2, x]^T to [6, 0, x]^T?
11. (Original post by matte30)
For this question; yes. But what if the unit you're comparing with a 0? example v is changed from [6, 2, x]^T to [6, 0, x]^T?
Then it's not solveable. Your scalar multiple would have to be zero. And with that x would be 0, and there's no way you can get 6 by multiplying something by zero.
12. (Original post by ghostwalker)
Then it's not solveable. Your scalar multiple would have to be zero. And with that x would be 0, and there's no way you can get 6 by multiplying something by zero.
Thank you very much; you've been very helpful

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