Question

If three non-zero vectors are \( \vec{a}, \vec{b} \) and \( \vec{c} \) such that
\( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \) and \( \vec{a} \times \vec{b} = \vec{a} \times \vec{c} \),
then show that \( \vec{b} = \vec{c} \).

Answer 1

Step 1: Given,
\( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \Rightarrow \vec{a} \cdot (\vec{b} - \vec{c}) = 0 \)
So, \( \vec{a} \) is perpendicular to \( (\vec{b} - \vec{c}) \).

Step 2: Given,
\( \vec{a} \times \vec{b} = \vec{a} \times \vec{c} \Rightarrow \vec{a} \times (\vec{b} - \vec{c}) = \vec{0} \)
So, \( \vec{a} \) is parallel to \( (\vec{b} - \vec{c}) \).

Step 3:
So we have:
- \( \vec{b} - \vec{c} \) is perpendicular to \( \vec{a} \)
- \( \vec{b} - \vec{c} \) is also parallel to \( \vec{a} \)

The only vector that can be both perpendicular and parallel to the same non-zero vector is the zero vector.

Therefore,
\( \vec{b} - \vec{c} = \vec{0} \Rightarrow \vec{b} = \vec{c} \)