### Understanding Motion on a Parametric CurveAssume that an object is moving along a parametric curve, and the three vector functions $ \mathbf{T}(t) $, $ \mathbf{N}(t) $, and $ \mathbf{B}(t) $ all exist at a particular point on that curve. **Problem Statement:**Circle the ONE statement below that MUST BE TRUE:(a) $ \mathbf{B} \cdot \mathbf{T} = 1 $(b) $ \mathbf{T} \times \mathbf{B} = \mathbf{N} $ (B is the binormal vector.)(c) $ \mathbf{N}(t) = \frac{\mathbf{v}(t)}{|\mathbf{v}(t)|} $(d) $ \mathbf{N}(t) $ always points in the direction of velocity $ \mathbf{v}(t) $.(e) $ \mathbf{a}(t) $ lies in the same plane as $ \mathbf{T}(t) $ and $ \mathbf{N}(t) $.**Explanation of Terms:**- $ \mathbf{T}(t) $: The unit tangent vector to the curve at time $ t $.- $ \mathbf{N}(t) $: The unit normal vector, which is perpendicular to $ \mathbf{T}(t) $.- $ \mathbf{B}(t) $: The binormal vector, defined as $ \mathbf{T}(t) \times \mathbf{N}(t) $.- $ \mathbf{v}(t) $: The velocity vector of the object.- $ \mathbf{a}(t) $: The acceleration vector of the object.**Detailed Explanations:**1. **Statement (a):** $ \mathbf{B} \cdot \mathbf{T} = 1 $ - The dot product between the unit tangent vector and the binormal vector should be zero because they are perpendicular.2. **Statement (b):** $ \mathbf{T} \times \mathbf{B} = \mathbf{N} $ - This is correct as per the definition of the binormal vector and its orthogonality properties.3. **Statement (c):** \( \mathbf{N}(t) = \frac{\mathbf{v}(t

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Assume that an object is moving along a parametric curve and the three vector function T (t), N(t) , and B (t) all exist at a particular point on that curve. CIRCLE the ONE statement below that MUST BE TRUE: (a) B. T=1 (b) T x B=N (B is the binormal vector.) v (t) V (c) N (t) = I0) 시 (d) N (t) always points in the direction of velocity v (t). (e) a (t) lies in the same plane as T (t) and N (t).

Assume that an object is moving along a parametric curve and the three vector function T (t), N(t) , and B (t) all exist at a particular point on that curve. CIRCLE the ONE statement below that MUST BE TRUE: (a) B. T=1 (b) T x B=N (B is the binormal vector.) v (t) V (c) N (t) = I0) 시 (d) N (t) always points in the direction of velocity v (t). (e) a (t) lies in the same plane as T (t) and N (t).

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

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### Understanding Motion on a Parametric Curve

Assume that an object is moving along a parametric curve, and the three vector functions $ \mathbf{T}(t) $, $ \mathbf{N}(t) $, and $ \mathbf{B}(t) $ all exist at a particular point on that curve.

**Problem Statement:**
Circle the ONE statement below that MUST BE TRUE:

(a) $ \mathbf{B} \cdot \mathbf{T} = 1 $

(b) $ \mathbf{T} \times \mathbf{B} = \mathbf{N} $ (B is the binormal vector.)

(c) $ \mathbf{N}(t) = \frac{\mathbf{v}(t)}{|\mathbf{v}(t)|} $

(d) $ \mathbf{N}(t) $ always points in the direction of velocity $ \mathbf{v}(t) $.

(e) $ \mathbf{a}(t) $ lies in the same plane as $ \mathbf{T}(t) $ and $ \mathbf{N}(t) $.

**Explanation of Terms:**

- $ \mathbf{T}(t) $: The unit tangent vector to the curve at time $ t $.
- $ \mathbf{N}(t) $: The unit normal vector, which is perpendicular to $ \mathbf{T}(t) $.
- $ \mathbf{B}(t) $: The binormal vector, defined as $ \mathbf{T}(t) \times \mathbf{N}(t) $.
- $ \mathbf{v}(t) $: The velocity vector of the object.
- $ \mathbf{a}(t) $: The acceleration vector of the object.

**Detailed Explanations:**

1. **Statement (a):** $ \mathbf{B} \cdot \mathbf{T} = 1 $
- The dot product between the unit tangent vector and the binormal vector should be zero because they are perpendicular.

2. **Statement (b):** $ \mathbf{T} \times \mathbf{B} = \mathbf{N} $
- This is correct as per the definition of the binormal vector and its orthogonality properties.

3. **Statement (c):** \( \mathbf{N}(t) = \frac{\mathbf{v}(t

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