Calculus 3 vectors

The question is
If a=<a1,a2> and b=<b1,b2> are nonzero vectors that are not parallel, show that any other vector c in R²can be written as sa+tb, where s and t are suitably chosen scalers.
So this is a proof and I can solve it geometrically but I would love to solve it algebraically. I found help online where c=a+t(b-a) would give me the answer but I have NO IDEA where c=a+(b-a) came from! :s-smilie:

Reply 1

atsruser

11

Original post by jasonj0118

The question is
If a=<a1,a2> and b=<b1,b2> are nonzero vectors that are not parallel, show that any other vector c in R²can be written as sa+tb, where s and t are suitably chosen scalers.
So this is a proof and I can solve it geometrically but I would love to solve it algebraically. I found help online where c=a+t(b-a) would give me the answer but I have NO IDEA where c=a+(b-a) came from! :s-smilie:

1. Let

c=\begin{pmatrix} p \\ q \end{pmatrix}

and suppose that, as suggested,

\bold{c} = s\bold{a} + t \bold{b}

.

2. Express the relationship between the three vectors

\bold{a},\bold{b},\bold{c}

in matrix form.

3. Figure out what you have to be able to do to the matrix that arises in part 2 if you must be able to find the particular

s

and

t

for any vector

\bold{c}

.

4. What relationship must then hold between the components of

\bold{a}

and

\bold{b}

?

5. What does that relationship have to do with the non-parallel restriction?

Reply 2

BlueSam3

17

Alternatively:

1) Show that you can express

\begin{pmatrix} 1 \\ 0 \end{pmatrix}

and

\begin{pmatrix} 0 \\ 1\end{pmatrix}

in terms of

\bold{a}

and

\bold{b}

in the given way.
2) Note that you can express any vector

\bold{c} \in \mathbb{R}

in terms of

\begin{pmatrix} 0 \\ 1\end{pmatrix}

and

\begin{pmatrix} 1 \\ 0\end{pmatrix}

in a very obvious way.
3) Substitute your representations for

\begin{pmatrix} 0 \\ 1\end{pmatrix}

and

\begin{pmatrix} 1 \\ 0\end{pmatrix}

in terms of

\bold{a}

and

\bold{b}

to complete the proof.

If you want to go further, you could demonstrate that this representation is unique.

Calculus 3 vectors

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