Scalar product problem

D

damon354

New member
Joined
Dec 23, 2020
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Hello,
I am stuck on this problem:

A, B, C and D are four points such that:
[MATH]\overrightarrow{BA} [/MATH] ⋅ [MATH]\overrightarrow{BD}[/MATH] ≤ 0 and [MATH]\overrightarrow{DB} [/MATH] ⋅ [MATH]\overrightarrow{DC}[/MATH] ≤ 0
Prove that AC ≥ BD.

Thanks for your help!
 
Welcome to the forum!
You asked for help but failed to state what type of help you need. Remember that this is a math help forum so we do not solve problems for students. We prefer that the student solve their own problem with hints from helpers on the forum.
What do these dot products being negative tell you? How can a dot product be positive?
 
damon354 said:
A, B, C and D are four points such that:
[MATH]\overrightarrow{BA} [/MATH] ⋅ [MATH]\overrightarrow{BD}[/MATH] ≤ 0 and [MATH]\overrightarrow{DB} [/MATH] ⋅ [MATH]\overrightarrow{DC}[/MATH] ≤ 0
Prove that AC ≥ BD.
Click to expand...
Are you expected to use purely vector methods to prove it?

What have you tried?
 
damon354 said:
No it is an open problem. I you can give me a starting point or hints...
Click to expand...
What does "open problem" mean? I presume you mean there are no stated restrictions on how you do it; but that doesn't tell me what methods are actually available to you. What topics have you been learning? What theorems are available? Proofs always have context.

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In any case, I would first restate what is given in terms of whatever method you are interested in trying. You might define some vectors (e.g. u = AB, ...) and state the givens and the goal in those terms; or you might approach it geometrically by stating that angles ABD and BDC are obtuse. But do something, so we have a place to start.

In terms of vectors, you want to show that `|AC| >= |BD|`; my first thought is to aim at `|AC|^2 - |BD|^2 >= 0`, and express that in terms of dot products.
 
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