Introduction to Basic Vectors: A Beginner's Guide

[synkk]

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

 

[FeDeX_LaTeX]

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?

 

[berkeman]

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...

 

[synkk]

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?

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)

To get started, draw the figure and label all the points and vectors...

I've done this, thanks.

 

[synkk]

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?