Question

If \(\vec{N}(t)\) is a unit normal vector, what is \(\vec{N}(t) \cdot \vec{r}^{\prime}(t) ?\)

Short answer

\( \vec{N}(t) \cdot \vec{r}^{\prime}(t) = 0 \)

Step-by-step solution

Recall the Properties of a Unit Normal Vector

A unit normal vector, denoted as \( \vec{N}(t) \), is a vector of unit length (magnitude 1) that is perpendicular to the tangent vector of a curve at point \( t \). It is used to describe the "normal" direction of a curve in space.

Define the Tangent Vector of the Curve

The tangent vector to a curve is typically denoted by \( \vec{T}(t) \) and is calculated as \( \vec{T}(t) = \frac{\vec{r}^{\prime}(t)}{\|\vec{r}^{\prime}(t)\|} \), where \( \vec{r}^{\prime}(t) \) is the derivative of the position vector concerning \( t \). It represents the instantaneous direction of the curve at \( t \).

Understand the Dot Product of Perpendicular Vectors

The dot product \( \vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos(\theta) \), where \( \theta \) is the angle between the vectors. If two vectors are perpendicular to each other, the dot product of these vectors is zero because \( \cos(90^\circ) = 0 \).

Identify the Relationship Between \( \vec{N}(t) \) and \( \vec{r}^{\prime}(t) \)

Since \( \vec{N}(t) \) is a unit normal vector, by definition, it is perpendicular to the tangent vector \( \vec{T}(t) \), and consequently \( \vec{r}^{\prime}(t) \). This implies \( \vec{N}(t) \cdot \vec{r}^{\prime}(t) = 0 \) due to the perpendicularity.

Key concepts

Unit Normal Vector

A unit normal vector is like the arrow that shows us which way is "up" or "down" with respect to a curve. It's always perpendicular to the curve's path at any given point. Imagine traveling along a winding road; the unit normal vector tells us which direction is directly away from the surface of the road. This vector is defined to always have a length of 1, ensuring it's only about direction, not size. In vector calculus, we often denote this vector as \( \vec{N}(t) \), where \(t\) represents a specific point on the curve. Its main job is to provide a sense of "normalcy" by being orthogonal to the tangent vector which describes the curve's direction.

Tangent Vector

The tangent vector is crucial for understanding how a curve changes direction at any point. It's like placing a mini-arrow along the curve pointing in the direction you're moving. Mathematically, we get this vector by taking the derivative of the position vector, denoted as \( \vec{r}^{\prime}(t) \). To ensure this vector only tells us about direction, we normalize it to get \( \vec{T}(t) \), making its length one. This approach helps highlight exactly how the curve twists and turns through space.

Dot Product

The dot product is a powerful tool for figuring out relationships between two vectors. Suppose we've got vectors \( \vec{a} \) and \( \vec{b} \). Their dot product is calculated as \( \vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos(\theta) \), where \(\theta\) is the angle between the vectors. Think of it as a way of measuring how much one vector "projects" onto another. Importantly, if two vectors are perpendicular, such as a unit normal vector and a tangent vector, their dot product is zero because \( \cos(90^\circ) = 0 \). This feature helps us quickly confirm when two directions are indeed at right angles in space.