How Do I Express Vector PA as a Linear Combination of Vectors a and b?

[MSG100]

The Problem:
Let O be the origin and let A, B, C be three points so that the quadrilateral OABC makes an parallelogram.
Name (1/4){OA} a, and the diagonal {OB} b. Let P be the point that splits the side OC in the ratio 3 :2 from O.
Write the vector {PA} as a linear combination of a and b.


Attempt to the soulution:
I have drawn the parallelogram and made some circumlocutions of the vectors.

How shall I tackle this problem?

 

[PeroK]

Can you say anything about AB and OC?

 

[haruspex]

You've written AB = (3/2)(PO). The sign is wrong (did you mean OP?) and the ratio is wrong.
The 2/3 in the next line is also wrong. P splits OC in the ratio 3:2, not 2:1.

 

[MSG100]

Yes, AB = (3/2)(PO) is wrong. It should be AB=-0C and therefore (2/3)AB=-(2/3)0C

(2/3)0C=-P0

Is it correct?

It is 3:2 ratio but it maybe looks like 2:1. I have split it in three parts and 2 of them are P0. I hope that's right.

 

[haruspex]

Yes, AB = (3/2)(PO) is wrong. It should be AB=-0C and therefore (2/3)AB=-(2/3)0C

(2/3)0C=-P0

Is it correct?

It is 3:2 ratio but it maybe looks like 2:1. I have split it in three parts and 2 of them are P0. I hope that's right.

You're missing my point about the ratio. If I have a line of length 15 and cut it into two pieces in the ratio 3:2, what are the two lengths?

 

[MSG100]

So embarrassing, of course it should 5 pieces.
3:2 of 15 is 9 and 6.

Thanks for notice this stupid mistake!

 

[MSG100]

Here's my 2nd attempt.

I'm not sure if it's right at all.

 

[haruspex]

Here's my 2nd attempt.

I'm not sure if it's right at all.


View attachment 64358

Looks like the right answer, but you have a sign error in the third line, which goes away in the fifth line. Maybe a transcription error?

 

[MSG100]

Thanks for your answer! Which sign do you mean? Is it when I change from AB=-0C to 0C=-AB

That's the thing I'm not sure of is if AB=-0C is correct? Isn't the vector in the direction from C to 0 (C0) and therefore AB=C0 or one can write AB=-0C?

 

[haruspex]

Thanks for your answer! Which sign do you mean? Is it when I change from AB=-0C to 0C=-AB

That's the thing I'm not sure of is if AB=-0C is correct? Isn't the vector in the direction from C to 0 (C0) and therefore AB=C0 or one can write AB=-0C?

Your third line starts OP = -(3/5)AB. Why the minus?
But where you use that to substitute for OP in the 4th line to get the 5th line, you seem to have used it without the minus sign.

 

[MSG100]

Maybe I'm blind but the minus is there all the time. In the third line: -(3/5)AB and in the fifth line -(3/5)(0B-4*0Q) and these two the same thing because I just substitute AB with (0B-4*0Q). Correct me if I'm wrong.

I think of 0P as (3/5)AB in the opposite direction and that's why I have the minus sign. Maybe I shall see AB and 0C as the same value?

 

[haruspex]

Maybe I'm blind but the minus is there all the time. In the third line: -(3/5)AB and in the fifth line -(3/5)(0B-4*0Q) and these two the same thing because I just substitute AB with (0B-4*0Q). Correct me if I'm wrong.

I think of 0P as (3/5)AB in the opposite direction and that's why I have the minus sign. Maybe I shall see AB and 0C as the same value?

Yes, the mistake starts one line earlier than I noticed: AB=OC, not -OC.
Having written (wrongly) OP = -(3/5)AB and (correctly) PA = -OP+4OQ, the logical deduction is PA = -(-(3/5)AB)+4OQ = (3/5)AB+4OQ = (3/5)(OB-4OQ)+4OQ. But you wrote -(3/5)(OB-4OQ)+4OQ, thereby correcting the earlier sign error.

 

[MSG100]

Okay I can see my mistake but I'm still confused.
(3/5)(0B-4OQ)+40Q and -(3/5)(0B-4OQ)+40Q doesn't give the same answer.

(3/5)(OB-4OQ)+4OQ= (3/5)0B+(8/5)0Q
and
-(3/5)(OB-4OQ)+4OQ=-(3/5)0B+(32/5)0Q

 

[haruspex]

Okay I can see my mistake but I'm still confused.
(3/5)(0B-4OQ)+40Q and -(3/5)(0B-4OQ)+40Q doesn't give the same answer.

(3/5)(OB-4OQ)+4OQ= (3/5)0B+(8/5)0Q
and
-(3/5)(OB-4OQ)+4OQ=-(3/5)0B+(32/5)0Q

You made two mistakes that cancelled.
The correct line 2, AB=OC, leads to what you posted, PA=-(3/5)(OB-4OQ)+4OQ.
Your version, AB=-OC, should have led you to PA=(3/5)(OB-4OQ)+4OQ, but your second mistake happened to yield the correct line PA=-(3/5)(OB-4OQ)+4OQ.

 

[MSG100]

Now I understand. It reminds me of the musical joke:
If you play a wrong note once it's a mistake. But if you play the wrong note twice, it's jazz.

Two mistake and I got the answer right!So I hope this is correct?

 

[haruspex]

Now I understand. It reminds me of the musical joke:
If you play a wrong note once it's a mistake. But if you play the wrong note twice, it's jazz.

Two mistake and I got the answer right!


So I hope this is correct?

View attachment 64367

Looks good.

 

[MSG100]

Thank you for the support and patience!