- #1
synkk
- 216
- 0
I can't even get started on part i), if anyone could give me a starting point and see where I go from there... thanks
Introduction to Basic Vectors: A Beginner's Guide
- Thread starter synkk
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- Vectors
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Homework Help Overview
The discussion revolves around a geometry problem involving vectors and the proof of a theorem related to the medians of a triangle. Participants are exploring the relationships between points and vectors in the context of this theorem.
Discussion Character
- Exploratory, Conceptual clarification, Mathematical reasoning
Approaches and Questions Raised
- Participants suggest starting with a diagram to visualize the problem. There is a focus on proving that the medians intersect in a specific ratio and questions arise about how to approach this proof using vectors rather than coordinates. Additionally, there are inquiries regarding the interpretation of variables in a related part of the problem.
Discussion Status
The discussion is ongoing, with some participants providing guidance on initial steps, such as drawing figures and labeling vectors. There is an acknowledgment of the need to prove the 2:1 ratio of the median segments, and participants are actively seeking clarification on the problem's requirements.
Contextual Notes
Some participants express uncertainty about how to construct proofs using vectors, indicating a potential gap in understanding the application of vector concepts in this context. There is also mention of specific variables and their meanings, which may require further exploration.
I can't even get started on part i), if anyone could give me a starting point and see where I go from there... thanks
- #2
FeDeX_LaTeX
Science Advisor
- 436
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I would start by drawing a picture. In part (i), they are asking for a proof of a very popular theorem in geometry; that each of the medians (the lines drawn from each vertex to the mid-point of the opposite side) intersect such that the median is split into two segments in the ratio 2:1 (i.e. the intersection is two thirds of the length of the median away from the vertex). Can you prove this?
- #3
berkeman
Admin
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synkk said:
I can't even get started on part i), if anyone could give me a starting point and see where I go from there... thanks
To get started, draw the figure and label all the points and vectors...
- #4
synkk
- 216
- 0
FeDeX_LaTeX said:
no I can't construct a proof of this using vectors, I've constructed it using a coordinate system before but not with vectors. Using this fact I was able to prove i), but I'm not sure how I can prove the fact (which I'm sure I'll have to do)
berkeman said:
I've done this, thanks.
- #5
synkk
- 216
- 0
here is what I have for part i):
## \vec{OL} = \dfrac{1}{2}(\vec{OA} + \vec{OC}) ##
## \vec{OG} = \vec{OB} + \dfrac{2}{3} \vec{BL} ##
## \vec{BL} = \vec{OL} - \vec{OB} = \dfrac{1}{2} (\vec{OA} + \vec{OC}) - \vec{OB} ##
subbing this into ## \vec{OG} ## I get the required result
I think I need to prove that the medians are split in a 2:1 ratio, but how would I do it using vectors?
Also for part ii)what do they mean by ## a^2 + b^2 + c^2 ## it makes a note that ## a = |\vec{BC}| ## then what is b and c?
## \vec{OL} = \dfrac{1}{2}(\vec{OA} + \vec{OC}) ##
## \vec{OG} = \vec{OB} + \dfrac{2}{3} \vec{BL} ##
## \vec{BL} = \vec{OL} - \vec{OB} = \dfrac{1}{2} (\vec{OA} + \vec{OC}) - \vec{OB} ##
subbing this into ## \vec{OG} ## I get the required result
I think I need to prove that the medians are split in a 2:1 ratio, but how would I do it using vectors?
Also for part ii)what do they mean by ## a^2 + b^2 + c^2 ## it makes a note that ## a = |\vec{BC}| ## then what is b and c?
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