If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three non-zero unequal vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, then find the angle between $\vec{a}$ and $\vec{b} - \vec{c}$.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $90^\circ$ ##
Given that $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three non-zero unequal vectors.
The given condition is $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$.
Simplify it:
$\vec{a} \cdot \vec{b} - \vec{a} \cdot \vec{c} = 0$
$\vec{a} \cdot (\vec{b} - \vec{c}) = 0$
This implies either $\vec{b} = \vec{c}$ or $\vec{a}$ and $\vec{b} - \vec{c}$ are perpendicular to each other.
As the vectors are unequal, $\vec{a}$ and $\vec{b} - \vec{c}$ are perpendicular to each other.
Therefore, the angle between $\vec{a}$ and $\vec{b} - \vec{c}$ is $90^\circ$.