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

Question

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.

Answer 1

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.

Answer 2

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