Question

Good morning. :) I have three vectors. I'd like to express the third vector in terms of the the first two. My work so far is attached. Thanks!

Answer 1

0 i-1 i=a(3i+4j)+b(2i+3j)
0=3a+2b ...(1)
-1=4a+3b ...(2)
3*(1) -2*(2) gives
0+2=9a+6b-8a-6b
2=a
from (1)
0=3*2+2b
2b=-6
b=-3
a=2
b=-3
?

Answer 2

Ah, I think I see it now. Just a sec.

Answer 3

Ok, your way makes sense, but what does my book mean when it shows the proccess (attached). I've added the i/j hat. Is there a way I'm supposed to approach it with matrices?

Answer 4

Your approach is ok, but you made mistake at: arranging the matrix
let see
\(a\left[\begin{matrix}3\\4\end{matrix}\right]+b\left[\begin{matrix}2\\3\end{matrix}\right]=\left[\begin{matrix}0\\-1\end{matrix}\right]\)

Answer 5

that is 3a+2b = 0
4a +3b =-1

Answer 6

Ah.

Answer 7

\(\left[\begin{matrix}3&2&|~~~~~0\\4&3&|-1\end{matrix}\right]\)

Answer 8

Ok, so from there I should go to reduced row echelon? Or is that only if they both start = 0?

Answer 9

rref, please

Answer 10

very easy,
divide both sides of row1 by 3

Answer 11

Yeah, I can get to the same answer, I just want to be sure I'm on a solid foundation conceptually.

Answer 12

yes, you are. :)

Answer 13

Ok, that's it! Not sure why my brain is taking so long to warm up this morning. Perhaps it's time for another cup of tea. :) Thank you for your help. Now I just need to practice for speed.