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My target is to solve next task: We have an triangle with points $A,B,C$. $AC$ => $\vec{\text{a}}$, $CB$ => $\vec{\text{b}}$ and $AB$ is divided by 6 equals parts with points $O, P, Q, R, S, T$. We must find width of $\vec{\text{CO}}$, $\vec{\text{CP}}$, $\vec{\text{CQ}}$, $\vec{\text{CR}}$, $\vec{\text{CS}}$ and $\vec{\text{CT}}$.

I know how to find vector $\vec{\text{CQ}}$, which is the middle of $AB$, by formula $\vec{\text{CQ}} = \frac{1}{2}(\vec{\text{a}} + \vec{\text{b}})$, but I do not know how to find others. Please explain. Thank you.

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If segment $AB$ has to be divided in six equal parts, point $T$ has to be $B$ ($T=B)$ You can use the formula:

$$\vec a+(n/6)(\vec b-\vec a)$$

It works for every point in $AB$ E.g., with n=3 we get Q:

$$\vec{\text{CQ}}=\vec a+(3/6)(\vec b-\vec a)=(1/2)(\vec b+\vec a)$$

With $n=5$, $S$:

$$\vec{\text{CS}}=\vec a+(5/6)(\vec b-\vec a)$$

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    Nope. T is not B – 2017-02-09
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    if $\vec{\text{a}} = 4$ and $\vec{\text{b}} = 4$ how much will be $\vec{\text{CS}}$? – 2017-02-09
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    Let $k$ the length of segment $AB$, we have $|AO|=k/6$, $|AP|=2k/6$, $|AQ|=3k/6=k/2$... $|AT|=6k/6=k$ You are equating in your comment vectors to numbers. If you are talking about lengths, in this case, trigonometry is needed. – 2017-02-09
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    My question: if if $\vec{\text{a}} = 4$ and $\vec{\text{b}} = 4$, $\vec{\text{CQ}} = \frac{1}{2}(4 +4) => 4$, correct? and how much will be $\vec{\text{CS}}$ by your formula it has to be $\vec{\text{CS}}=\vec{\text{a}}+ \frac{5}{6}(4-4)$ => ? – 2017-02-09
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    No, wrong. You *can't* equate a vector to a number, it's no sense. – 2017-02-09
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    Is it proper if I say that $\vec{\text{CQ}}$ = $\frac{3}{6}(\vec{\text{a}}+\vec{\text{b}})$? – 2017-02-09
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    and $\vec{\text{CT}}$ = $\frac{5}{6}(\vec{\text{a}}+\vec{\text{b}})$? – 2017-02-09
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    All expressions are now well-formed, but the last is false. It's $\vec{\text{CT}}=\vec b$ because $T=B$ – 2017-02-09
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    I dissaggree. Ti s not B. B is last point of AB. T is point before B – 2017-02-09
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    Then your segment is split into 7 parts and not 6. The reasoning given by @RafaBudría is still valid (with minor changes) – 2017-02-09