If \( \vec{a}.\vec{b}=\ \vec{a}.\vec{c} \) and \( \vec{a}\times\ \vec{b}=\ \vec{a}\times\ \vec{c} \) , \( \vec{a}\ne\ \vec{0} \), then
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \vec{b}-\ \vec{c}\ =\ \vec{0} \)
Since \( \vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c} \) and \( \vec{a}\times\vec{b}=\vec{a}\times\vec{c} \),
\( \displaystyle \vec{a}\cdot(\vec{b}-\vec{c})=0 \)
\( \displaystyle \vec{a}\times(\vec{b}-\vec{c})=\vec{0} \)
The first says \( \vec{b}-\vec{c} \) is perpendicular to \( \vec{a} \) (or is zero).
The second says \( \vec{b}-\vec{c} \) is parallel to \( \vec{a} \) (or is zero).
A non-zero vector cannot be both perpendicular and parallel to the non-zero vector \( \vec{a} \). So \( \vec{b}-\vec{c}=\vec{0} \).