Multiple Choice
Which statement is true about the relationship between points and planes in three-dimensional space?
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8. Vectors
Geometric Vectors
Multiple Choice
Given two pairs of vectors, and , and and , if = and = , 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.
Verified step by step guidance1
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.
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
\[
\|\mathbf{e}\| \|\mathbf{f}\| \cos(\theta_{ef}) = 0
\]
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:
\[
\cos(\theta_{ab}) = 0 \quad \text{and} \quad \cos(\theta_{ef}) = 0
\]
Recall that \( \cos(\theta) = 0 \) when \( \theta = 90^\circ \) or \( \theta = 270^\circ \), meaning the vectors are perpendicular (orthogonal) to each other.
Therefore, the conclusion is that each pair of vectors \( (\mathbf{a}, \mathbf{b}) \) and \( (\mathbf{e}, \mathbf{f}) \) are perpendicular to each other.
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