Question

True or False? In Exercises 67-70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a particle moves along a sphere centered at the origin, then its derivative vector is always tangent to the sphere.

Short answer

The statement is TRUE. The derivative vector of a particle moving along a sphere is indeed always tangent to the sphere provided that the sphere is centered at the origin and the particle is moving just on the surface of the sphere.

Step-by-step solution

Understanding the Statement

Visualize a sphere. If a particle is moving on the surface of this sphere and at any moment we take the derivative of its position vector, it means we determine the velocity of the particle. This velocity vector is also called tangent vector. The statement is claiming that this vector is always tangent to the sphere.

Verifying the statement

By the nature of spheres, and by the definition of tangents, we should find that the velocity vector of a particle moving along the sphere at any point is indeed tangent to the sphere. This is because the movement of the particle follows the surface of the sphere and not the interior or exterior. Visualization or drawing may assist in understanding this concept. So, this statement seems to be true. However, it's crucial to understand that the derivative vector should point towards the direction of the motion of the particle.

Final Resolution

The derivative vector of a particle moving along a sphere is indeed always tangent to the sphere, but this is true only if the sphere centered at the origin and the particle isn't moving in or out, but just on the surface. Therefore, the statement 'If a particle moves along a sphere centered at the origin, then its derivative vector is always tangent to the sphere.' is TRUE.

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This mathematical framework is essential for understanding changes between values that are related by a function. One of the fundamental operations in calculus is differentiation, which calculates the rate at which a quantity changes. In practical terms, if we consider a particle moving through space, calculus allows us to determine its velocity at any given moment by deriving its position function with respect to time.

Understanding the derivative of a position vector leads to insights about the motion of objects, such as a particle moving along the surface of a sphere. Applying calculus to this scenario provides a deeper comprehension of the principles governing the particle's motion and the vector forces in play, which are directly related to the concepts of tangent and velocity vectors.

Tangent Vector

A tangent vector represents the direction of motion of an object moving along a curve at a specific point and is perpendicular to the radius at that point in the case of a circle or sphere. This concept extends to multiple dimensions. For a particle traveling on the surface of a sphere, visualizing the tangent vector is helpful: it is the arrow that skims the sphere's surface at the point of contact without cutting in or bulging out. In calculus, determining this vector involves taking the derivative of the particle's position with respect to time.

The significance of the tangent vector lies in its ability to describe the exact direction of the object's motion, and it is a powerful tool in understanding the geometry and dynamics of curved spaces, such as the sphere in our exercise.

Velocity Vector

The velocity vector is a specific example of a tangent vector that describes the speed and direction of an object's motion. In the case of a particle moving on a sphere, the velocity vector is indeed a tangent vector because it shows the instantaneous direction of travel along the sphere's surface. It is essential to note that the vector itself represents velocity, so its magnitude is equal to the speed of the particle, while its direction is tangent to the sphere.

The velocity vector is an integral part of analyzing dynamic systems in physics and engineering, providing information about how fast an object is moving and where it is going at any point in time. In essence, it captures the essence of the particle's journey around the sphere.

Derivatives

In mathematics, derivatives represent how a function changes as its input changes. They are a measure of the rate of change or the slope of a curve at any given point. When we derive the position vector of a moving particle with respect to time, we obtain its velocity vector. This operation applies to our spherical example, where we describe a particle's movement over time and want to know the direction and speed—the velocity—at which it is traveling.

In the context of our exercise, it's important to grasp that the truthfulness of the statement hinges on the understanding of derivatives. Simply put, for a particle on a sphere centered at the origin, the derivative of the position vector is indeed always tangent to the surface of the sphere, and this explains why the particle never 'dives' into or 'jumps' out of the sphere as it moves. This phenomenon is a direct consequence of the mathematical principles surrounding derivatives.