If the resultant of $\vec{A}$ and $\vec{B}$ is $\perp$ (perpendicular) to $\vec{A}$, then it means that
- $B\sin\theta = |\vec{A}|$
- $R = B\cos\theta$
So angle b|w (between) $\vec{A}$ and $\vec{B} = 90^\circ + 60^\circ = 150^\circ$ given that $R = B/2 \implies B/2 = B\cos\theta \implies \cos\theta = 1/2 \implies \theta = 60^\circ$
How is $B\sin\theta$ equal to the magnitude of $\vec{A}$ and $B\cos\theta = R$?
$B\cos\theta$ can be big or smaller than $R$, so how can we say that they have same value?
Please explain.
This is not stated explicitly, but $\vec R$ must be defined as
$\vec R = \vec A + \vec B$
Since we know that $\vec R$ is perpendicular to $\vec A$, then $|\vec A|$ and $|\vec R|$ are perpendicular components of the vector $\vec B$. Since the angle between $\vec B$ and $\vec R$ is $\theta$, we have
$|\vec R| = |\vec B| \cos \theta$
and
$|\vec A| = |\vec B| \sin \theta$