Further vectors

[waxen mortar]

I somehow got infinite solutions for I and J

[long ridge]

You should have 8z from 2 lots of equation 1

But

I don’t think this works for finding the equation of the plane

You should probably just use vector product on AB and AC to find the normal to the plane and then use the form r.n = a.n and expand it out

[waxen mortar]

Ph yes ur right

[waxen mortar]

Ohh I see

Thank you

[waxen mortar]

We haven’t covered this part of the unit yet but what is r.n and a.n

Like what is r, a and n

Well assuming n is normal

[long ridge]

So r is a general point on the plane so (x,y,z), n is the normal vector

And a is any position on the plane

So that could be A,B or C

[waxen mortar]

I see thanks so much

[long ridge]

Using the vector product method is just easier imo the other algebra is pretty tedious as far as I can remember

[waxen mortar]

Is that when u form a 3x3 matrix of top row ijk then find the determinant

[long ridge]

Yeah

[waxen mortar]

Oh okay thanks, how would I do this thoughr, would my 2nd and 3rd row just be any of ABC?

[long ridge]

So you know that any two vectors linking 2 of the 3 points together will span the plane

Well I should say non collinear but it doesn’t really make sense if they are ig

So their vector product produces a vector that’s normal to the plane

So you want to choose any 2 of AB, BC, AC

[waxen mortar]

Thank you so much I’ll try this and I’ll let yk how I get on 👍

[waxen mortar]

I got a and b wrong but c is right

For a it should be x+y=2

And for b it should just be (1,1,0) which the mark scheme has as the normal vector

[long ridge]

Why did you dot (-2,2,-2) with (1,-1,4)

You have the normal vector (-10,-10,0) so
-10x -10y = (3,-1,4) dot (-10,-10,0) so
-10x -10y = -30 +10
-10x -10y = -20 or x+y=2

[long ridge]

for b you can have multiple answers btw

Since it can be any multiple of (1,1,0)

[waxen mortar]

I thought that’s how u find d with the other 2 (b.c) but now I’m realising cos u explained I should’ve dot r.n

[waxen mortar]

Thank you

[waxen mortar]

But I’m confused isn’t 1,1,0 just the direction vector, would u have to let lambda = 1 for instance then do
r=a + lambdaB

[long ridge]

1,1,0 is a vector normal to the surface of the plane

So is 2,2,0

Or any multiple of that vector since all you’re doing is changing the length of said vector

Its direction is constant

[waxen mortar]

Ohh okay thank you so much