Question

Using vectors and dot product show the diagonals of a parallelogram have equal lengths if and only if it’s a rectangle.

The proof is given on P.2 of: http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/part-a-vectors-determinants-and-planes/session-3-uses-of-the-dot-product-lengths-and-angles/MIT18_02SC_we_5_comb.pdf

However, there are 2 parts in the proof, which I don't quite understand:

1) Why is BC+CD=BC-AB?

2) Why is |AC|^2=|BD|^2.
i.e. why is |AB|^2+2AB*BC+|BC|^2=|BC|^-2BC*AB+|AB|^2

Answer 1

these are vectors.
BC - AB = BC + BA if it is a rectangle, BA = CD

Answer 2

BC+CD=BC-AB? @irkiz

Answer 3

*do you mean BC+CD=BC-AB?

Answer 4

oh wait I'm beginning to see what you're saying

Answer 5

-BA = BA

Answer 6

yes, got you

Answer 7

you mean -AB=BA?
or -BA=AB is valid too?

Answer 8

*-BA=BA i mean

Answer 9

-BA is not equal to BA
this is vectors.
- means opposite direction
means different vectors

Answer 10

yes, understood. so only -BA=AB and -AB=BA

Answer 11

-BA = AB but -BA not= BA
however |-BA| = |BA|
yup

Answer 12

understood

Answer 13

the second part is a bit hard to explain on a computer. give me a moment to try to figure out how to explain

Answer 14

I think I understand how the second part follows from the first. It's just geometry and vectors again right?

Answer 15

yup

Answer 16

same principle as in first part?

Answer 17

yes

Answer 18

the only thing im still a bit confused about is the 4AB*BC=0 in the end

Answer 19

is it just |AC|^2-|BD|^2 multiplied out?

Answer 20

the problem consists of two parts:
(i) the parallelogram is a rectangle IF AC = BD; and
(ii) AC = BD if the parallelogram is a rectangle

(i) assuming AC = BD we prove that the angles are right angles (the dot product equals zero)

Answer 21

AC.AC = AB.AB + 2 AB.BC + BC.BC
BD.BD = BC.BC - 2 AB.BC + AB.AB
because segment AC = segment BD it follows that AC.AC = BD.BD

therefore AB.AB + 2 AB.BC + BC.BC = BC.BC - 2 AB.BC + AB.AB

cancelling equal terms
2 AB.BC = -2 AB.BC

or, after transposing the term on the RHS
4 AB.BC = 0
therefore AB.BC =0
AB is perpendicular to BC
hence the parallelogram is a rectangle

Answer 22

@nexis i hope i made that part clear to you.

Answer 23

@sirm3d thinking through your explanation right now

Answer 24

yes that was absolute sense.

Answer 25

makes*

Answer 26

Thank you :)

Answer 27

for the long and involved explanation. Just starting out with multi-variable, still need to get used to thinking in vectors

Answer 28

(ii) part two of the proof:
the reverse solution begins with AB.BC = 0 (the angle formed is a right angle)
and ends in AC.AC = BD.BD (the diagonals are congruent)

Answer 29

Yes, thank you :)