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### Vector Proof

Date: 01/17/99 at 00:42:34
From: Olivia
Subject: Vector proofs

I've been having some difficulty with solving vector proofs. I kind of
know how to do them but don't know the proper way to write the
solution. I was wondering if you could please give me some tips or
help me out with the questions. It would be much appreciated.

1. Given: P, Q, R, and S any 4 non-collinear points
A and B the midpoints of PR and QS respectively
Prove: vector PQ + vector RS = 2(vector AB)

2. Given parallelogram PQRS with A dividing vector SR in the ratio
3:4, B and C the points of trisection of vector QR, and T the
point of intersection of vector SB, show that BT:TS = 4:9

_____________________ S
\                  . \ 3
\                . / \ A
\            / .     \
\      /    .        \ 4
\/_______.___________\
Q        B      C      R

There is supposed to be a line from B to S and another from Q to A.

Date: 01/17/99 at 16:37:38
From: Doctor Anthony
Subject: Re: Vector proofs

Question 1:

Take origin at A, and let b, p, q, r, s be the position vectors of B,
P, Q, R, S relative to the origin A.

We have r = -p and  b = (q+s)/2

Then:

PQ = q-p    RS = s-r

PQ + RS = q-p + s-r
= q+s - p + p    since r = -p
= q+s
= 2b
= 2 AB     as required.

Question 2:

Take origin at R and let s, q, be the vectors RS ans RQ respectively.
Any point on the line QA is q + k(4s/7 - q) = q(1-k) + s(4k/7) where k
is a scalar.

Any point on the line SB is s + k'(2q/3 - s) =  q(2k'/3) + s(1-k')
where k' is a scalar.

Where these lines intersect at T we can equate coefficients of q and s
for the two lines.

We get the two equations:

Coeffs of q   1-k = 2k'/3        (1)

Coeffs of s   4k/7 = 1-k'        (2)

From (1) k' = 3(1-k)/2 and putting this in (2) we get:

4k/7 = 1 - 3(1-k)/2
8k = 14 - 21(1-k)
8k = 14 - 21 + 21k
7 = 13k
k = 7/13

From this:

k' = 3(1-k)/2 = 3(6/13)/2
k' =  9/13

So ST/SB = 9/13, and therefore ST/TB = 9/4 or BT:TS = 4:9, as required.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Linear Algebra

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