Question

So if I am given a 4D vector, how do I tell what kind of region it forms? For example a line, or a plane?
U = <2R, -R, 4R, R>

Answer 1

@Abdulhameed, @Ballplop, @cwrw238, @DavidUsa

Answer 2

Man if only i knew. I cant help you with this one sorry.

Answer 3

It's ok, thanks.

Answer 4

@iambatman

Answer 5

Help me Batman (;

Answer 6

4 dimensional

Answer 7

a 1D vector is a line, 2D vector is a line, 3D vector is a line

i think a 4D vector must also be a line

Answer 8

I think when we break this vector into each of its component it can form a plane or something else?

Answer 9

That being said, for every line there is a perpendicular plane passing through the origin. If you wish, you may interpret the vector as that plane.

Answer 10

No wait that's in three dimensions. In four there might be a perpendicular 3D 'plane'.

Answer 11

<a, b, c, d> may be interpreted as the equation ax + by + cz + dw = 0. Assuming d is nonzero, for every x, y, z there is a number w that satisfies the equation, namely w = -(ax + by + cz)/d, so it looks three dimensional.

Answer 12

Since I can break this vector into a set of 4 vectors that forms a basis for R4....

Answer 13

Oh, now I get your question.

Answer 14

Sorry I didn't word it very well in the beginning.

Answer 15

So you want to know what the set of all linear combinations of \((2R, -R, 4R, R)\) looks like. Every such combination is of the form \(2R\mu_1 - R\mu_2 + 4R\mu_3 + R\mu_4 = R(2\mu_1 - \mu_2 + 4\mu_3 + \mu_4)\) which is a scalar multiple of R. So it looks like a line. Did I get it right this time?

Answer 16

Yes I believe so.

Answer 17

Does this have anything to do with that the set of vectors <2, 0, 0, 0>, <0, -1, 0, 0>, <0, 0, 4, 0>, <0, 0, 0, 1> ?

Answer 18

No not really.

Answer 19

Thank you so much for all of your help!

Answer 20

You're welcome!

Answer 21

a vector in 2D , 3D or 4D , etc
can be thought of as picking out a single point.
if you have
U = <2R, -R, 4R, R>
and R is an arbitrary scalar (is it?) this is U= R<2,-1,4,1>
and as you change R you lengthen (or shorten) U... along a line in the direction of U