Multiple Choice
In physics, how does velocity differ from speed?
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3. Vectors
Review of Vectors vs. Scalars
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If two vectors are not perpendicular to each other, which of the following statements is true about their dot product ()?
A
Their dot product () is always negative.
B
Their dot product () is zero.
C
Their dot product () is nonzero unless one of the vectors is the zero vector ().
D
Their dot product () is always equal to the product of their magnitudes ().
Verified step by step guidance1
Recall the definition of the dot product between two vectors \( \mathbf{a} \) and \( \mathbf{b} \):
\[
\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \times |\mathbf{b}| \times \cos(\theta)
\]
where \( \theta \) is the angle between the two vectors.
Understand that if the vectors are not perpendicular, then the angle \( \theta \) is not 90 degrees, so \( \cos(\theta) \neq 0 \). This means the dot product is not zero unless one of the vectors has zero magnitude.
Analyze the sign of the dot product: since \( \cos(\theta) \) can be positive or negative depending on the angle (less than 90° or greater than 90°), the dot product is not always negative; it can be positive or negative.
Note that the dot product equals the product of the magnitudes only when \( \cos(\theta) = 1 \), i.e., when the vectors are parallel and pointing in the same direction, which is not guaranteed if they are just non-perpendicular.
Conclude that the correct general statement is: the dot product is nonzero unless one of the vectors is the zero vector.
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