Let $u$ be a unit vector, and let $v$ and $w$ be vectors, where $w$ is also a unit vector. Which of the following statements is true about the dot products $u\cdot v$ and $u\cdot w$?

• A. The value of $u\cdot v$ is always less than $0$.
• B. The value of $u\cdot w$ is always greater than $1$.
• C. The value of $u\cdot v$ is always equal to $1$.
• D. The value of $u\cdot w$ is always between $-1$ and $1$.

Answer: (D)

Verified step by step guidance

1. Recall the definition of the dot product between two vectors \( \mathbf{a} \) and \( \mathbf{b} \): \[ \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta \] where \( \theta \) is the angle between the vectors.
2. Since \( \mathbf{u} \) and \( \mathbf{w} \) are unit vectors, their magnitudes are both 1, so the dot product simplifies to: \[ \mathbf{u} \cdot \mathbf{w} = 1 \times 1 \times \cos \theta = \cos \theta \]
3. The cosine of an angle \( \theta \) always lies between -1 and 1, inclusive. Therefore, the value of \( \mathbf{u} \cdot \mathbf{w} \) must be between -1 and 1.
4. For the dot product \( \mathbf{u} \cdot \mathbf{v} \), since \( \mathbf{v} \) is not necessarily a unit vector, the value can vary widely depending on the magnitude of \( \mathbf{v} \) and the angle between \( \mathbf{u} \) and \( \mathbf{v} \). It is not restricted to any fixed range like the unit vector case.
5. Therefore, the only always true statement is that the dot product of two unit vectors \( \mathbf{u} \cdot \mathbf{w} \) lies between -1 and 1.