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Chapter 6: Problem 1

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

Short Answer

Expert verified
Two non-zero vectors are orthogonal, or at 90 degrees to each other, if their scalar product is zero.

Step by step solution

01

Define Scalar Product

Firstly we define the scalar product (also known as dot product) of two vectors. If we have \( \vec{a} \) and \( \vec{b} \) are two vectors, their scalar product \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \), where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \).
02

Analyze Scenario

In this problem, it's given that dot product of the two vectors \( \vec{a} \cdot \vec{b} = 0 \). Substituting the definition from above, we get \( |\vec{a}| |\vec{b}| \cos(\theta) = 0 \).
03

Reason the Solution

The magnitude of a nonzero vector is always positive, so the product of the magnitudes of two nonzero vectors \( |\vec{a}| |\vec{b}| \) will also be positive. For this equation to be 0, the cos(\theta) must be 0. The cosine is 0 only for \( \theta = 90 degrees \) or \( \theta = 270 degrees \). Therefore, the two vectors must be orthogonal (or perpendicular) to each other if their scalar product is zero.

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