How to find $x$ and $y$ components of a vector if angle between two vectors is Obtuse (greater than 90)?

Question

How to divide $Q$ and $P$ into its components?
Can you write $θ$ in terms of P and Q. if yes then how to write it and from which equation you are calculating the θ?

Answer 1

Take the horizontal (cosine) component of the acute angle with the X-axis, the direction of the vector will be opposite to that of P. However you can also take cosine with the obtuse angle, but you will get a negative value indicating that the direction of vector is not along P

Answer 2

Assume that the the point of intersection of $\vec{Q}$ and $\vec{P}$ is the origin of the plane. Because $\vec{Q}$ has horizontal component exactly $-\vec{P}$ and vertical component exactly $\vec{R}$, then $\vec{Q} = \vec{R}-\vec{P}$. Since $\vec{P}$ has no component in the vertical direction, then it is already divided into horizontal and vertical components.

Then recall that $$\frac{\vec{Q}\cdot\vec{P}}{|\vec{Q}||\vec{P}|}=\cos(\theta)$$ so $$\cos^{-1}\left(\frac{\vec{Q}\cdot\vec{P}}{|\vec{Q}||\vec{P}|}\right)=\theta$$