Math Is Fun Forum

!nval!d_us3rnam3

Member

Registered: 2017-03-18

Posts: 46

Vector problem, help ASAP

(a) Show that any two-dimensional vector can be expressed in the form

where

and

are real numbers.

(b) Let

and

be non-zero vectors. Show that any two-dimensional vector can be expressed in the form:

where

and

are real numbers, if and only if of the vectors

and

, one vector is not a scalar multiple of the other vector.

I have no idea how to start; can you work me through how to prove it?

---

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Vector problem, help ASAP

Hi !nval!d_us3rnam3,

Thanks for your post -- I fixed your LaTeX.

For part (a), suppose you've got some vector

in

. Then, you've just got to solve this pair of simultaneous equations for

and

:

Does that make sense? (Let me know if anything sounds confusing -- happy to help.)

For part (b), suppose that instead of

and

you have

and

. This gives you the system:

What sorts of conditions do you need here for that to have a solution?

!nval!d_us3rnam3

Member

Registered: 2017-03-18

Posts: 46

Re: Vector problem, help ASAP

For part (a), how do I prove this equation is the one I should use?
Also, for part (b), I don't understand what conditions you mean.

---

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Vector problem, help ASAP

No worries -- let's look at part (a) first.

(a) Show that any two-dimensional vector can be expressed in the form

where

and

are real numbers.

Okay, let's take any two-dimensional vector, say,

. We want to know: can we find real numbers

and

so that:

If we 'multiply out' the left-hand side, we get:

Now, we can add two vectors just by adding the matching components, so that:

In other words, we want to find real numbers

and

so that:

which is exactly the same as solving the pair of simultaneous equations:

Remember, we're solving for

and

here. (Just pretend that

and

are any old real numbers.)

Let me know if this makes sense -- happy to explain anything further if you need more help.

!nval!d_us3rnam3

Member

Registered: 2017-03-18

Posts: 46

Re: Vector problem, help ASAP

I think I have part (a), but can you explain part (b) in a little more detail?
Thanks in advance!

---

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

!nval!d_us3rnam3

Member

Registered: 2017-03-18

Posts: 46

Re: Vector problem, help ASAP

Bump... can anyone else walk me through part (b)? I don't understand the scalar multiple bit.

Last edited by !nval!d_us3rnam3 (2019-05-23 07:43:17)

---

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Vector problem, help ASAP

Part (b) is similar, but bear in mind that you have to prove both directions. First, let's see if we can do the reverse direction (not scalar multiples => any vector can be represented in the form

).

Suppose that instead of

and

you have

and

. In other words, instead of having

you have this instead:

Another way of writing this is:

Now, let's say you wanted to solve this system for

and

. You'd need to find the inverse of

, right? But for that inverse to exist, the determinant can't be equal to 0. In other words, for that thing to have a solution, you must have:

i.e.

which tells you that

isn't a multiple of

. Remember, saying that

and

are scalar multiples of each other just means that you can find some real number

such that

, or in other words:

i.e.

Now, for the forwards direction (any vector can be expressed as

=>

and

aren't multiples of each other), try proving the contrapositive, i.e. show that if

for some real number

, then not every vector can be expressed in the form

. Let me know if you have any more questions!