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Chapter 6: Problem 2

If the scalar product of two nonzero vectors is zero, what can you conclude about their relative directions?

Short Answer

Expert verified
If the scalar product of two nonzero vectors is zero, they are perpendicular to each other.

Step by step solution

01

Definition of Scalar Product

The scalar, or dot, product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by \( \vec{a} \cdot \vec{b} = ||\vec{a}|| \, ||\vec{b}|| \, \cos{\theta} \), where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \), and \( ||\vec{a}|| \) and \( ||\vec{b}|| \) denote the magnitudes of the vectors.
02

Scalar Product Equals Zero

If the scalar product of \( \vec{a} \) and \( \vec{b} \) is zero, i.e. \( \vec{a} \cdot \vec{b} = 0 \), then \( ||\vec{a}|| \, ||\vec{b}|| \, \cos{\theta} = 0 \). Since \( \vec{a} \) and \( \vec{b} \) are nonzero vectors, their magnitudes are not zero. Therefore, the only way for their scalar product to be zero is if \( \cos{\theta} = 0 \).
03

Conclusion on Relative Directions

The cosine function equals zero at \( \theta = 90^\circ \) or \( \theta = 270^\circ \). However, because the angle between two vectors is always less than or equal to \( 180^\circ \) in Euclidean space, we can conclude that the two vectors are perpendicular to each other when their scalar product is zero.

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