VCE Specialist 3/4 Question Thread!

[BubbleWrapMan]

They could also be collinear, but in that case they're coplanar as well

If one or more of the vectors in the set is the zero vector then they are automatically dependent, since the zero vector lies on every plane and every line

[zvezda]

But it's not necessarily true that ALL linear dependent vectors are coplanar, or even collinear for that matter. Am I correct in saying this? The definition of coplanar vectors is where the vectors share the same plane isnt it? That is, for example the first quadrant of the cartesian plane?

[BubbleWrapMan]

Quadrant doesn't matter - three vectors are linearly dependent if they are all on the same plane. Doesn't have to be one of the x-y, y-z, or x-z planes - it can be any plane. By that logic, the quadrant is irrelevant too - a plane extends infinitely in every direction, so three vectors can be dependent if they're simply on the same plane, regardless of what quadrant they're in.

Three vectors have to be coplanar (or 'less', e.g. collinear, or some of them are zero) in order to be dependent. If they are not all in the same plane then they must be independent. Note that I'm only talking about the specific case of three vectors in three dimensions, here.

[nspire]

Can somebody explain to me in simple terms how a test for linear dependence/independence could be done efficiently using a CAS? I believe it involves matrices and determinant but I was never actually taught it in class

[BubbleWrapMan]

Type all the vectors into a matrix, with each vector as a row (the matrix doesn't necessarily need to be square)

Use the 'rref' function on that matrix

The vectors are dependent if and only if the resulting matrix has one or more zero rows



For the specific case of n vectors in n dimensions, you can type the vectors into a square matrix, and find the determinant, which will be zero if and only if the vectors are dependent

[nspire]

Both ways seem pretty simple, thanks!

[BubbleWrapMan]

Just make sure you only do it for multiple-choice questions

It pains me to say that they probably wouldn't accept these methods in written-response questions (since learning extra stuff in VCE is bad)

[zvezda]

So not all independent vectors are coplanar?

[BubbleWrapMan]

You mean dependent?

If you consider collinearity to be a degenerate case of coplanarity, and the zero vector to lie on every plane, then any three 3D vectors are dependent if and only if they are coplanar

A set of four or more 3D vectors is always dependent, regardless of whether they're coplanar

A set of four 4D vectors could possibly be dependent without being coplanar

A set of two vectors in any dimensions are dependent if and only if they are parallel (if you consider 0 to be parallel to every vector)


It's more useful to consider the actual definition of linear dependence, since all of these cases are simply a result of it. Attempting to memorise these cases won't be too useful.

[zvezda]

You mean dependent?

If you consider collinearity to be a degenerate case of coplanarity, and the zero vector to lie on every plane, then any three 3D vectors are dependent if and only if they are coplanar

A set of four or more 3D vectors is always dependent, regardless of whether they're coplanar

A set of four 4D vectors could possibly be dependent without being coplanar

A set of two vectors in any dimensions are dependent if and only if they are parallel (if you consider 0 to be parallel to every vector)


It's more useful to consider the actual definition of linear dependence, since all of these cases are simply a result of it. Attempting to memorise these cases won't be too useful.

I do know the significance of linear dependence, I'm just trying to figure out why, in Derrick Ha's book, it says "three linearly dependent vectors are always coplanar". So, by the definition of a plane being basically the whole sort of area where graphs and such are plotted, then that does make sense. But, going by what I believed before, where a plane was a quadrant, this wasnt necessarily true to me.

[brightsky]

that's basically the definition of linear dependence:

two 2D vectors are linearly dependent iff they are parallel.

three 3D vectors are linearly dependent iff they are coplanar.

let's say you have three random linearly dependent vectors. this means that we can get back to where we started by re-arranging the vectors and/or making them shorter or longer. test it out: this is only possible if the three vectors are coplanar (lie on the same plane). 

[jono88]

if z1=root2 +root2 i and z2=-root+root2 i and are roots of a quadratic equation. Find the equation in the form of az^2+bz+c, where a,b and c are real numbers.

i have b as -2root2 i which isn't a real number :/ Is there something wron

[jono88]

*z2=-root2 + root2 i

[brightsky]

the factors are (z - z1) and (z - z2) yeah? so the quadratic equation will be of the form:

a(z-z1)(z-z2) = 0
just expand and simplify.

[jono88]

yes i did that and i got z^2- (2root2)iz-4