1. ## Vector Proofs

Ive tried this problem numerous times but im constantly going in circles. Not looking for the answer because that would defeat the purpose of doing this question, more for just a small hint if possible

ABCD is a rectangle. Prove OA · OC = OB · OD
where OA, OC, OB and OD are all vectors.

point O is an origin NOT on the same plane as the rectangle. any help is appreciated. thanks

2. Originally Posted by Finroe
Ive tried this problem numerous times but im constantly going in circles. Not looking for the answer because that would defeat the purpose of doing this question, more for just a small hint if possible

ABCD is a rectangle. Prove OA &#183; OC = OB &#183; OD
where OA, OC, OB and OD are all vectors.

point O is an origin NOT on the same plane as the rectangle. any help is appreciated. thanks
Although this might not help you, I would recommend thinking of it like a pyramid with a rectangular base, and O is the top of that pyramid (vertex).

Just to get a visual...

3. Originally Posted by Finroe
Ive tried this problem numerous times but im constantly going in circles. Not looking for the answer because that would defeat the purpose of doing this question, more for just a small hint if possible

ABCD is a rectangle. Prove OA · OC = OB · OD
where OA, OC, OB and OD are all vectors.

point O is an origin NOT on the same plane as the rectangle. any help is appreciated. thanks
Hello,

as you requested: Only a hint:

$\displaystyle \overrightarrow{OA}=\overrightarrow{OB}+\overright arrow{BA}$

$\displaystyle \overrightarrow{OC}=\overrightarrow{OB}+\overright arrow{BC}$

Multiply the RHSs of these equations. One of the summands is zero because you have to do with a rectangle with orthogonal sides(?). You can factor the sum. That's it

EB

4. First off, thanks for the prompt replies. Ive used the hints and theyve been very helpful. i overlooked the fact that two perpindicular vectors have a dot product of 0, our dot product unit was a while ago. I was just wondering, if anyone has tried solving this problem and if they have, how long is their solution. My proof is close to 3/4 of a page and i still havent proved the equality. let me know if you managed to solve this and how logn the proof was. thanks again.

5. Originally Posted by Finroe
First off, thanks for the prompt replies. Ive used the hints and theyve been very helpful. i overlooked the fact that two perpindicular vectors have a dot product of 0, our dot product unit was a while ago. I was just wondering, if anyone has tried solving this problem and if they have, how long is their solution. My proof is close to 3/4 of a page and i still havent proved the equality. let me know if you managed to solve this and how logn the proof was. thanks again.
Hello,

I repeat my hint:

$\displaystyle \overrightarrow{OA}=\overrightarrow{OB}+\overright arrow{BA}$

$\displaystyle \overrightarrow{OC}=\overrightarrow{OB}+\overright arrow{BC}$

According to your problem you'll get the equuation:

$\displaystyle \overrightarrow{OA} \cdot \overrightarrow{OC}=\left( \overrightarrow{OB}+\overrightarrow{BA} \right) \cdot \left( \overrightarrow{OB}+\overrightarrow{BC} \right)$

Expand the RHS:

$\displaystyle \overrightarrow{OA} \cdot \overrightarrow{OC}=\overrightarrow{OB} \cdot \overrightarrow{OB}+\overrightarrow{OB} \cdot \overrightarrow{BC} +\overrightarrow{BA} \cdot \overrightarrow{OB} +\underbrace{\overrightarrow{BA} \cdot \overrightarrow{BC}}_{\text{equals zero}}$

Factor the RHS:

$\displaystyle \overrightarrow{OA} \cdot \overrightarrow{OC}=\overrightarrow{OB} \cdot \underbrace{\left( \overrightarrow{OB} + \overrightarrow{BC} + \overrightarrow{BA} \right)}_{equals\ \overrightarrow{OD}}$

Notice please that with vectors the following equation is true:
$\displaystyle \overrightarrow{BA}=\overrightarrow{CD}$

Thus:

$\displaystyle \overrightarrow{OA} \cdot \overrightarrow{OC}=\overrightarrow{OB} \cdot \overrightarrow{OD}$

EB