How to Express Vectors of a Regular Hexagon in Terms of Given Components?

[anemone]

ABCDEF is a regular hexagon with $\vec {BC}$ represents $\underline {b}$ and $\vec {FC}$ represents 2$\underline {a}$. Express, vector
$\vec {AB}$, $\vec {CD}$ and $\vec {EC}$ in terms of $\underline {a}$ and $\underline {b}$.

Before I start, I want to ask if we need to redefined $\underline {b}$ and 2$\underline {a}$? I mean, let $\underline {b}$ as

$$\begin{pmatrix} m \\0 \end{pmatrix}$$.

Thanks.

 

[Evgeny.Makarov]

You don't have to express $\underline{a}$ and $\underline{b}$ using their coordinates, though it's possible to get the answer that way as well. Suppose $\underline{a}=(k,l)$ and $\underline{b}=(m,0)$ and you express, say, $\vec{AB}$ using k, l and m. The difficulty is that you need express $\vec{AB}$ as a combination of specifically $(k,l)$ and $(m,0)$, not just as some expression of k, l and m.

It is clear that $\vec{EC}=\vec{FB}=\vec{FC}-\vec{BC}=2\underline{a}-\underline{b}$. To get a geometric intuition about the regular hexagon you can also look at this page.

 

[soroban]

Hello, anemone!

I don't understand the notation $\underline{b}$.
If $\underline{b}$ represents $\overrightarrow{BC}$, isn't $b$ also a vector?

$ABCDEF\text{ is a regular hexagon with }\vec{b} = \overrightarrow {BC}\text{ and }2\vec{a} = \overrightarrow{FC} $

$\text{Express vectors }\overrightarrow{AB},\;\overrightarrow{CD},\; \overrightarrow{EC}\text{ in terms of }\vec{a}\text{ and }\vec{b}.$

          A       B
          * - - - *
         / \     / \
        /   \   /   \ b
       /  a  \ /  a  \
    F * - - - * - - - * C
       \     / \     /
        \   /   \   /
         \ /     \ /
          * - - - *
          E       D

$\overrightarrow{AB} \:=\:\vec{a}$

$\overrightarrow{CD} \:=\:\vec{b} - \vec{a}$

$\overrightarrow{EC} \:=\:2\vec{a} - \vec{b}$

 

[anemone]

You don't have to express $\underline{a}$ and $\underline{b}$ using their coordinates, though it's possible to get the answer that way as well. Suppose $\underline{a}=(k,l)$ and $\underline{b}=(m,0)$ and you express, say, $\vec{AB}$ using k, l and m. The difficulty is that you need express $\vec{AB}$ as a combination of specifically $(k,l)$ and $(m,0)$, not just as some expression of k, l and m.

It is clear that $\vec{EC}=\vec{FB}=\vec{FC}-\vec{BC}=2\underline{a}-\underline{b}$. To get a geometric intuition about the regular hexagon you can also look at this page.

Got it. Thanks, Evgeny.Makarov.
Maybe I'm just trying too hard...and not knowing that I'm actually trying to complicate the simple problem.

---------- Post added at 06:05 AM ---------- Previous post was at 05:59 AM ----------

I don't understand the notation $\underline{b}$.
If $\underline{b}$ represents $\overrightarrow{BC}$, isn't $b$ also a vector?

I had the same reaction as you when I first read the problem!
Anyway, thanks, Soroban.
Now I fully understand with the help of the diagram and it really is as simple as that.:)

 

[CellCoree]

Hello,

Thank you for your question. In this context, it is not necessary to redefine $\underline {b}$ and 2$\underline {a}$. The notation $\underline {b}$ represents a vector in the direction of $\vec {BC}$ and 2$\underline {a}$ represents a vector in the direction of $\vec {FC}$ with twice the magnitude of $\underline {a}$. Therefore, we can express the vectors $\vec {AB}$, $\vec {CD}$, and $\vec {EC}$ as follows:

$$\vec {AB} = \underline {b} - \underline {a}$$
$$\vec {CD} = 2\underline {a}$$
$$\vec {EC} = \underline {b} + \underline {a}$$

I hope this helps clarify the notation and express the vectors in terms of $\underline {a}$ and $\underline {b}$. Let me know if you have any further questions.

Best,