Give a parameterization of the circle of radius a centered...

Give a parameterization of the circle of radius $a$ centered at the point $\left(x_0, y_0, z_0\right)$ and in the plane parallel to two given unit vectors $\vec{u}$ and $\vec{v}$ such that $\vec{u} \cdot \vec{v}=0$.

Key Concepts

Translation in Euclidean Space

Translation in Euclidean space involves shifting every point of a geometric object by the same constant vector. In the parameterization of a circle, after describing the circle centered at the origin, translating by the center coordinates moves the circle to the desired location in space, ensuring that the circle is centered at the given point.

Orthonormal Basis of a Plane

When two unit vectors are orthogonal (their dot product is zero), they form an orthonormal basis for a plane. This means they are perfectly perpendicular and each has a magnitude of one, which simplifies the process of defining directions in the plane. With such a basis, any vector in the plane can be expressed as a linear combination of these two unit vectors.

Parameterization of Curves

Parameterization is the process of representing a curve by expressing its coordinates as functions of one or more parameters. In the context of a circle, it typically involves using trigonometric functions so that each coordinate can be smoothly traced as the angle parameter varies, which allows for a complete description of all the points on the circle.

Circle Geometry

A circle in the plane is described by all the points at a fixed distance (radius) from a given center. Using trigonometric functions (like cosine and sine), one can express the coordinates of any point on the circle relative to the center. This is critical for providing a complete and continuous representation of the circular path.

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