Demonstration of vector components projected to a new axis pair. The new axes are rotated by an angle [math]\alpha[/math].[br]The steps for computing the components on the new axes using the step button (Double right triangles)[br]1. The original x and y vector components[br]2. Show new axes pair and angle alpha[br]3. Project the y component to the new x axis, [math]V_{yx}=V_y\cdot sin\left(\alpha\right)[/math][br]4. Project the y component to the new y axis, [math]V_{yy}=V_y\cdot cos\left(\alpha\right)[/math][br]5. Project the x component to the new x axis, [math]V_{xx}=V_x\cdot cos\left(\alpha\right)[/math][br]6. Project the x component to the new y axis, [math]V_{xy}=-V_x\cdot sin\left(\alpha\right)[/math][br][br]The final operation is to add the new components on each axis.[br][math]V_{x_{new}}=V_{xx}+V_{yx}=V_x\cdot cos\left(\alpha\right)+V_y\cdot sin\left(\alpha\right)[/math][br][math]V_{y_{new}}=V_{xy}+V_{yy}=-V_x\cdot sin\left(\alpha\right)+V_y\cdot cos\left(\alpha\right)[/math][br][br]Note that these two equation can be written in matrix form

Another method is to use the magnitude and angles of the vectors.[br]The magnitude of the original vector, [math]\left|V\right|=\sqrt{\left(V_x\right)^2+\left(V_y\right)^2}[/math][br]and the angle of the original vector using the arctangent [math]\beta=tan^{-1}\left(\frac{V_y}{V_x}\right)[/math] adding or subtracting 180[sup]o[/sup] if [i]V[sub]x[/sub][/i] is negative.[br]Then the angle of the vector with respect to the new coordinates is [math]\gamma=\beta-\alpha[/math] and the magnitude stays the same. Therefor the new components become [math]V_{x_{new}}=\left|V\right|cos\left(\gamma\right)[/math] and [math]V_{y_{new}}=\left|V\right|sin\left(\gamma\right)[/math]