How Do You Calculate Specific Points in a Triangle Using Vectors?

[bjgawp]

Homework Statement

Consider the triangle ABC whose vertices have position vectors a, b, and c respectively.

http://img106.imageshack.us/img106/4240/vectorat6.png

Find the position vector of
(a) P, the mid-point of AB
(b) Q, the point of trisection of AB, with Q closer to B.
(c) R, the mid-point of the median CP.

Homework Equations

None really. It just seems that I have a major conflict with the textbook answer and I'm not quite sure why. The textbook says:
(a) ½(b + a)
(b) ⅓(2b + a)
(c) ½(a + b + c)

The Attempt at a Solution

(a) I got this answer. Looking at the diagram, if we theoretically add OB and OA tip-to-tail, the half of the resultant vector should give us OP.
(b) Similarly to (a),
AB = b + a
OQ = ⅓AB
--> OQ = ⅓(b + a)
I don't see why b is multiplied by 2.

(c) First off:
CP = CO + OP
CP = -c + ½(b + a)

Then,
OR = OC + ½CP
OR = c + ½(-c + ½(b + a))
OR = ½c + ¼b + ¼a

Yeah, I don't see why my answers disagree with the textbook.

 

[Fermat]

a) OB = OA + AB
b = a + AB
therefore,
AB = b - a, not b + a

OP = OA + ½(AB) = a + ½(b - a) = a + ½b - ½a
OP = ½(b + a)

b) OQ = OA + (2/3)AB (Q is 2/3 of way along AB)

c) the book is wrong, you are right!

 

[Architeuthis Dux]

Can someone help me out here?


Hello,

Thank you for providing your attempted solutions and explaining your reasoning. I can see that you have a good understanding of vector addition and how to find mid-points and trisection points.

For part (b), the textbook answer of ⅓(2b + a) is correct. This is because when we add 2b and a, we are essentially adding b twice. This is because b is the vector from O to B, so adding it twice represents moving along b from O to B, and then moving along b again from B to C. Therefore, ⅓(2b + a) represents the point that is ⅓ of the way along the line AB, closer to B.

For part (c), I can see where your confusion lies. Your method of finding the position vector of R is correct, but the textbook answer is simplified further. Let's break it down:

OR = ½c + ¼b + ¼a

Since R is the midpoint of CP, we know that CR = ½CP. So we can rewrite this as:

OR = ½c + ¼(2c + b + a)

Now, we can factor out a ½ from the second term:

OR = ½c + ½(2c + b + a)

And we can see that the second term is just the vector from O to B, which is b + a. So we can write:

OR = ½c + ½(b + a)

Which is the same as the textbook answer of ½(a + b + c).

I hope this helps clarify the textbook answers for you. Keep up the good work with your vector problems!