Question

The resultant of two vectors \( \vec{A} \) and \( \vec{B} \) is perpendicular to \( \vec{A} \) and its magnitude is half that of \( \vec{B} \). The angle between vectors \( \vec{A} \) and \( \vec{B} \) is ________ .

Answer 1

Correct Answer: 150

The resultant vector $\vec{R}$ of $\vec{A}$ and $\vec{B}$ is perpendicular to $\vec{A}$. The magnitude of $\vec{R}$ is given as: 
\[ |\vec{R}| = \frac{|\vec{B}|}{2}. \] 

Using the vector projection formula, the component of $\vec{B}$ along $\vec{A}$ is: 
\[ B \cos \theta = \frac{B}{2}. \] 
Simplify: \[ \cos \theta = \frac{1}{2}. \] 

From this, $\theta = 60^\circ$. Since $\vec{R}$ is perpendicular to $\vec{A}$, the angle between $\vec{A}$ and $\vec{B}$ is: 
\[ \text{Angle between } \vec{A} \text{ and } \vec{B} = 90^\circ + 60^\circ = 150^\circ. \]