# The sum of the angle between two vectors and their intermediate vector is greater than the total angle

I have two vectors $$\vec{A}(2.735, -4.737)$$ and $$\vec{B}(2.735, 4.737)$$, and a vector $$\vec{P}(1.380, -0.796)$$ in between these two vectors. When I calculate the angle between $$\vec{A}$$ and $$\vec{B}$$ it is $$120$$ degrees. However, the angle between $$\vec{A}$$ and $$\vec P$$ is $$60$$ degrees and that of $$\vec P$$ and $$\vec B$$ is $$90$$ degrees. So the sum of individual angles $$(150)$$ is greater than the total angle. Why is this anomaly? Here is a plot of the three vectors $$\vec A,\vec B$$ and $$\vec P$$

• The angle between $\vec{A}$ and $\vec{P}$ is (approximately) $30°.$ Mar 9, 2023 at 11:43
The angle between $$\vec{A}$$ and $$\vec{P}$$ is \begin{align}\arccos\left(\frac{\vec{A}\cdot\vec{P}}{\left\|\vec{A}\right\|\left\|\vec{P}\right\|}\right)&=\arccos\left(\frac{2.735\cdot1.38+(-4.737)(-0.796)}{\sqrt{2.735^2+(-4.737)^2}\sqrt{1.38^2+(-0.796)^2}}\right)\\ &=\arccos(0.86583\ldots)\\ &=30^\circ\end{align} so you just made a calculation error.