Given two pairs of vectors, $a$ and $b$, and $e$ and $f$, if $a\cdot b$ = $0$ and $e\cdot f$ = $0$, what can be concluded about the relationship between each pair of vectors?

• A. Each pair of vectors is parallel to each other.
• B. Each pair of vectors is perpendicular to each other.
• C. Each pair of vectors points in the same direction.
• D. Each pair of vectors has the same magnitude.

Answer: (B)

Verified step by step guidance

1. Recall the definition of the dot product between two vectors \( \mathbf{u} \) and \( \mathbf{v} \): \[ \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos(\theta) \] where \( \theta \) is the angle between the two vectors.
2. Given that \( \mathbf{a} \cdot \mathbf{b} = 0 \) and \( \mathbf{e} \cdot \mathbf{f} = 0 \), substitute into the dot product formula: \[ \|\mathbf{a}\| \|\mathbf{b}\| \cos(\theta_{ab}) = 0 \] and \[ \|\mathbf{e}\| \|\mathbf{f}\| \cos(\theta_{ef}) = 0 \]
3. Since the magnitudes \( \|\mathbf{a}\|, \|\mathbf{b}\|, \|\mathbf{e}\|, \|\mathbf{f}\| \) are generally non-zero (unless vectors are zero vectors), the only way for the dot product to be zero is if: \[ \cos(\theta_{ab}) = 0 \quad \text{and} \quad \cos(\theta_{ef}) = 0 \]
4. Recall that \( \cos(\theta) = 0 \) when \( \theta = 90^\circ \) or \( \theta = 270^\circ \), meaning the vectors are perpendicular (orthogonal) to each other.
5. Therefore, the conclusion is that each pair of vectors \( (\mathbf{a}, \mathbf{b}) \) and \( (\mathbf{e}, \mathbf{f}) \) are perpendicular to each other.