If \(\vec a\cdot\vec b=\vec a\cdot\vec c, \vec a\times\vec b=\vec a\times\vec c\) and \(\vec a \ne 0\), then the vector \(\vec b\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\vec c\)
Let \(\vec d = \vec b - \vec c\). The two given equations become
\(\vec a \cdot \vec d = 0\)
\(\vec a \times \vec d = \vec 0\)
The dot product being zero means \(\vec d\) is perpendicular to \(\vec a\) (or \(\vec d = \vec 0\)). The cross product being zero means \(\vec d\) is parallel to \(\vec a\) (or \(\vec d = \vec 0\)). Since \(\vec a \ne \vec 0\), a vector cannot be both perpendicular and parallel to it unless it is the zero vector:
\(\vec b - \vec c = \vec 0\)
\(\vec b = \vec c\)
Tip: Neither condition alone allows cancelling \(\vec a\): \(\vec a \cdot \vec b = \vec a \cdot \vec c\) only gives \(\vec b - \vec c \perp \vec a\), and \(\vec a \times \vec b = \vec a \times \vec c\) only gives \(\vec b - \vec c \parallel \vec a\). Both together are needed.