Let \(\mathbf{n}_{\mathrm{t}}\) be a unit vector of variable direction with an origin at the coordinate origin. Let the position vector \(\mathbf{r}\) from the origin be determined by the equation where \(\mathbf{S}\) is a symmetric dyadic. When \(\mathbf{n}_{1}\) varies in direction, its terminus describes a unit sphere. Show that the terminus of \(\mathbf{r}\) then describes an ellipsoid. Hint: Assume the coordinate system is oriented such that \(\mathbf{S}\) is diagonal.

In the context of mathematics and vector calculus, a symmetric dyadic is a type of tensor that can be represented by a square matrix. More specifically, a dyadic tensor can be understood as a combination of two vectors, or in this case, unit vectors. When the dyadic is termed symmetric, it means that the matrix representing this tensor is equal to its transpose.

This characteristic is particularly important in physics and engineering, where symmetric dyadics often arise when considering the stress, strain, or inertia of materials and objects under various conditions. In terms of the matrix, symmetry implies that the off-diagonal elements are equal, i.e., the value in the i-th row and j-th column is the same as in the j-th row and i-th column. However, for our problem, we are dealing with a symmetric dyadic that is also diagonal, meaning that all off-diagonal entries are zero. This simplifies many equations and operations involving the tensor, and it's central to understanding how this leads to the equation for an ellipsoid.

The advantage of having the dyadic in a diagonal form is that each axis of the coordinate system is treated independently, avoiding the complications of cross-product terms. This simplification allows us to directly observe the influence of changes in one direction, isolated from the others.

A position vector, denoted usually as \(\textbf{r}\), indicates the position of a point in space relative to a fixed origin. In a three-dimensional Cartesian coordinate system, the vector is typically described by three coordinates \(x\), \(y\), and \(z\).

The concept of a position vector is central to understanding many geometrical shapes in vector space, including lines, planes, and—as seen in the given exercise—an ellipsoid. A position vector is not only used to describe the location of a point but also to traverse space and translate shapes. In our problem, the position vector is used in combination with a symmetric dyadic to transform a sphere into an ellipsoid. The transformation is achieved by scaling the components of the position vector by the entries of the diagonal dyadic.

Understanding how position vectors work is also fundamental to mastering concepts in mechanics, electromagnetism, and other physical sciences, where the position of particles and bodies in space is crucial.

A unit vector is a vector that has a magnitude of one and is used to indicate direction only. Unit vectors are often used to represent the basis of a coordinate system. For instance, the standard unit vectors in three dimensions are \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\), pointing in the directions of the x-axis, y-axis, and z-axis respectively.

In the exercise at hand, the variable unit vector \(\mathbf{n}_{t}\) represents all possible directions emanating from the origin. When this vector's direction changes, its tip describes the surface of a unit sphere, implying it remains at a constant distance (one unit) from the origin regardless of its orientation. The significance of the unit vector in our problem is profound; it is scaled by the symmetric dyadic, which distorts the unit sphere into an ellipsoid, reflected by the position vector \(\mathbf{r}\). This property shows how scaling a vector affects the geometry it is associated with and the shape it describes when its direction varies.

Unit vectors are foundational in vector algebra and calculus, as they simplify many problems by reducing them to issues concerning direction rather than magnitude. This simplification allows for understanding changes in geometry as in our example, going from a sphere to an ellipsoid, through directional manipulation.