Question

Find all planes parallel to plane passing through points (1,2,3),(1,2,7),(1,1,-3)

Answer 1

find the drs of the plane containing the points.
all the parallel planes will have same drs. only the thing which will differ is the constant term.

Answer 2

Parallel planes have the parallel normal vectors just different points. Recall a plane with surface normal \(\langle a,b,c\rangle\) and passing thru a point \(\langle x_0,y_0,z_0\rangle\) may be expressed using the equation \(\langle a,b,c\rangle\cdot\langle x,y,z\rangle=d\) where \(d=\langle a,b,c\rangle\cdot\langle x_0,y_0,z_0\rangle\). Parallel planes will just have different \(d\) :-)

Answer 3

@Abhishek619 is correct! @FutureMathProfessor you can find an appropriate normal vector by finding two vectors lying in your plane (how about difference vectors between those points?) and computing their cross product

Answer 4

I got normal vector <4,0,0>

Answer 5

@FutureMathProfessor correct, that works. So does \(\langle 1,0,0\rangle\)

Answer 6

So how does that help me now

Answer 7

@FutureMathProfessor recall what I wrote above. Given the normal vector \(\langle 1,0,0\rangle\) we may write:$$\langle1,0,0\rangle\cdot\langle x,y,z\rangle=d$$ for any real \(d\) to express a plane parallel to our own

Answer 8

trplets of the form (k,0,0) can be the drs of the required plane. provided k be a scalar.
meaning, all the planes are parallel to the y-z plane.

Answer 9

So just 4x = d is the answer..?

Answer 10

for all real \(d\) -- yes @FutureMathProfessor. Try sketching our original plane and a few of the form \(4x=d\) and it should be more intuitive

Answer 11

Don't I have to use 4(x-1) = d because of the offset in the points?

Answer 12

@FutureMathProfessor you don't have to offset anything; we have \(d=\langle a,b,c\rangle \cdot \langle x_0,y_0,z_0\rangle\) and considering all real \(d\) means we're considering planes passing through any point as long as they have the same surface normal.

Answer 13

That being said, \(4(x-1)=d\) is also a perfectly valid answer, since \(4(x-1)=4x-4=d\Longleftrightarrow 4x=d+4\) where \(d+4\) can also be any real number