Finding coordinates of vector with respect to new orthonormal basis

[RM5152]

Hello. I am stuck on this problem which I have attached a photo of. I have also attached a photo of my maths lesson relating to orthonormal basis and it’s quite limited so I’m not sure if I’m doing this right or wrong.. am I meant to just expand out the sum I have written so far in my work? Thanks for any help.

 

[blamocur]

Have you done 6.a and 6.b.i ?
Your solution for 6.b.ii is, at the very least, inefficient: did you notice what [imath]\mathbf v_1[/imath], [imath]\mathbf v_2[/imath] and [imath]\mathbf v_3[/imath] are in there?

 

[RM5152]

I know it’s inefficient , I just don’t know how to do it. I know the vectors are in there but I was going to try get help with the solution and how to do it before I started multiplying them all out because I imagine it will get very messy.

Have you done 6.a and 6.b.i ?
Your solution for 6.b.ii is, at the very least, inefficient: did you notice what [imath]\mathbf v_1[/imath], [imath]\mathbf v_2[/imath] and [imath]\mathbf v_3[/imath] are in there?

This is my answer to 6a and 6bi. The only direction I got on how to do these was the sheet I posted above so I simply have no idea how to do them. If you could help me out I would really appreciate it. I’ve never seen a vector written like this (av2 - bv3 + cv1) as opposed to (a,b,c) … I don’t know how to write it out, will the sum or entries not be incredibly long if I multiply PTx out like I have started to do in my workings?

 

[blamocur]

6.a and 6.a.i looks correct to me, but your "proofs" that [imath]||\mathbf v_k||=1[/imath] look too terse to me. I wonder if you teacher expects more details there.

I have a hint/leading question for 6.a.ii: what does it mean exactly to "determine the three coordinates of the given vector [imath]\mathbf x[/imath] with respect to [imath]\mathbf v_1, \mathbf v_2, \mathbf v_3[/imath] ?

 

[RM5152]

Could I please get a further hint on how to solve b(ii)? The only answer I have is that it means to determine how the vector would look in the new ‘on’ basis but this probably isn’t right. I would love some help on how to properly approach this. Thanks so much. ?

 

[blamocur]

Could I please get a further hint on how to solve b(ii)? The only answer I have is that it means to determine how the vector would look in the new ‘on’ basis but this probably isn’t right. I would love some help on how to properly approach this. Thanks so much. ?

But can you answer my question in the first hint? I'll make it even more direct: if you are told that the coordinates of [imath]\mathbf x[/imath] with respect to [imath]\mathbf v_1, \mathbf v_2, \mathbf v_3[/imath] are, for example, [imath]a,b,c[/imath] what does it mean by definition? I.e., can you express this statement with a simple formula?