# How to get coordinates of point knowing distance from x,y and angle?

I have such a problem :

I am given :

• x,y
• $\|a\|$
• $\alpha$
• $\vec{v}$ and $\|v\|$

I need to get the coordinates of point X1Y2.

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## 2 Answers

Use this fact that for two vector $v=(x_1,x_2),w=(y_1,y_2)$ we can evaluate $v.w$, the dot product of $v$ and $w$, by two ways. They are : $$v.w=x_1y_1+x_2y_2$$ and $$v.w=|v||w|\cos(\alpha)$$

Personally, I prefer @Karolis's answer but we can have an elementary approach according to what was given to us.

• $||a||=\sqrt{(X-X_1)^2-(Y-Y_2)^2}$
• $XX_1+YY_2=vw=||v||.||a||.\cos(\alpha)$

Above system have two equations of two unknowns. As you noted, we have $||a||,||v||,\alpha,X,Y$ so, put the known values and evaluate $X_1,Y_2$. I hope I could help.

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Thanks a lot for this but if I am correct I will have an equation with 2 unknowns, right ? – Patryk Jan 6 at 0:50
@Patryk: Right. But you have a system of equation with 2 unknowns and I think after solving this system, you will get X1 and Y1. Tell me if you have any problem in it. ;-) – Babak S. Jan 6 at 6:36
Somehow I can't see the second equation. Can you edit your answer so that it is a bit more clearer :) ? – Patryk Jan 6 at 10:45
@Patryk: Sorry and forgive me for the delay. – Babak S. Jan 6 at 21:02
Good job explaining! +1 – amWhy Feb 23 at 0:07

$$(x_1, y_1) = (x, y) + \frac{a}{\|v\|} \cdot R(\alpha) \cdot \vec{v}$$ Where $R(\alpha)$ is a rotation matrix.

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@Patryk: This is a best answer for a fast result.+1 ;-) – Babak S. Jan 6 at 11:00
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