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I have a point for example A (0,0) and B(10,10). Now I want to calculate a third point which lies in the same direction. I want to calculate point (x3,y3). I need a formula to calculate the new point. Please use (x0, y0) for (0,0) and (x1, y1) for (x1, y1) for answering. Thanks.

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Given two distinct points $A$ and $B$, you can use the idea of weighted average to immediately write down a parametric formula for the line through $A$ and $B$: $A(1-t)+Bt$. If $t=0$, you get $A$, if $t=1$, you get $B$, for other values of $t$ you get other points on that line.

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The line passing through points $(x_0, y_0)$ and $(x_1, y_1)$ has equation $$ y-y_0 = \big(\frac{y_1-y_0}{x_1-x_0}\big)(x-x_0) $$ which you can use to compute new values of $x$ or $y$.
You must be sure that $x_0$ is not equal to $x_1$.

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How we can use it to get third point? – Sami May 26 '14 at 12:50
@Sami, The values of $x_0$, $y_0$, $x_1$ and $y_1$ are known so my equation is really just in terms of $x$ and $y$. Choose a value for $x$, plug it in the equation and you will get the corresponding value of $y$. Alternatively, you can plug in a value of $y$ and find the corresponding $x$. – Peter Phipps May 26 '14 at 13:49

Let $\,\overrightarrow{AB}:= B-A=(10,10)$, so the line determined by these two points is, in vectorial parametric form, $\,A+t\overrightarrow{AB}=(10t,10t)\,,\,t\in\mathbb{R}$ .

Finally, if you want a third vector in the same direction from A to B just choose $\,t>0\,$

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