By a translation of axes, (distinct) points 𝐒(x1, y1), 𝐓(x2, y2), and 𝐔(x3, y3) are given new co

Step 1: Define the translation of axes. Let the translation vector be u005C(u005Cmathbf{v} u003D (h, k)u005C). T

By a translation of axes, (distinct) points 𝐒(x1, y1), 𝐓(x2,...

By a translation of axes, (distinct) points $\mathbf{S}\left(x_1, y_1\right), \mathbf{T}\left(x_2, y_2\right)$, and $\mathbf{U}\left(x_3, y_3\right)$ are given new coordinates $\mathbf{S}^{\prime}\left(x_1^{\prime}, y_1^{\prime}\right), \mathbf{T}^{\prime}\left(x_2^{\prime}, y_2^{\prime}\right)$, and $\mathbf{U}^{\prime}\left(x_3^{\prime}, y_3^{\prime}\right)$, respectively. If $\mathbf{s}, \mathbf{t}, \mathbf{u}, \mathbf{s}^{\prime}, \mathbf{t}^{\prime}$, and $\mathbf{u}^{\prime}$ are the vectors corresponding to $\mathbf{S}, \mathbf{T}, \mathbf{U}, \mathbf{S}^{\prime}, \mathbf{T}^{\prime}$, and $\mathbf{U}^{\prime}$, respectively, verify algebraically that $$ \frac{\left(\mathbf{t}^{\prime}-\mathbf{s}^{\prime}\right) \cdot\left(\mathbf{u}^{\prime}-\mathbf{s}^{\prime}\right)}{\left\|\mathbf{t}^{\prime}-\mathbf{s}^{\prime}\right\|\left\|\mathbf{u}^{\prime}-\mathbf{s}^{\prime}\right\|}=\frac{(\mathbf{t}-\mathbf{s}) \cdot(\mathbf{u}-\mathbf{s})}{\|\mathbf{t}-\mathbf{s}\|\|\mathbf{u}-\mathbf{s}\|} $$ and interpret this fact geometrically.

By a translation of axes, (distinct) points $\mathbf{S}\left(x_1, y_1\right), \mathbf{T}\left(x_2, y_2\right)$, and $\mathbf{U}\left(x_3, y_3\right)$ are given new coordinates $\mathbf{S}^{\prime}\left(x_1^{\prime}, y_1^{\prime}\right), \mathbf{T}^{\prime}\left(x_2^{\prime}, y_2^{\prime}\right)$, and $\mathbf{U}^{\prime}\left(x_3^{\prime}, y_3^{\prime}\right)$, respectively. If $\mathbf{s}, \mathbf{t}, \mathbf{u}, \mathbf{s}^{\prime}, \mathbf{t}^{\prime}$, and $\mathbf{u}^{\prime}$ are the vectors corresponding to $\mathbf{S}, \mathbf{T}, \mathbf{U}, \mathbf{S}^{\prime}, \mathbf{T}^{\prime}$, and $\mathbf{U}^{\prime}$, respectively, verify algebraically that

$$
\frac{\left(\mathbf{t}^{\prime}-\mathbf{s}^{\prime}\right) \cdot\left(\mathbf{u}^{\prime}-\mathbf{s}^{\prime}\right)}{\left\|\mathbf{t}^{\prime}-\mathbf{s}^{\prime}\right\|\left\|\mathbf{u}^{\prime}-\mathbf{s}^{\prime}\right\|}=\frac{(\mathbf{t}-\mathbf{s}) \cdot(\mathbf{u}-\mathbf{s})}{\|\mathbf{t}-\mathbf{s}\|\|\mathbf{u}-\mathbf{s}\|}
$$

and interpret this fact geometrically.

Key Concepts

Translation of Axes

This concept deals with shifting the entire coordinate system by a fixed vector. Such a translation does not alter the relative position of points (i.e., the differences between their coordinates remain the same), thereby leaving many geometric and algebraic properties, like distances and angles, invariant. This invariance is crucial in understanding how geometric figures behave under coordinate changes.

Vector Subtraction

Vector subtraction is used to determine the displacement between two points by subtracting their position vectors. Since the subtraction operation cancels out any common translation, the resulting displacement vector remains unaffected by translations of the coordinate system, preserving the geometric relationship between points.

Dot Product

The dot product of two vectors is an algebraic operation that multiplies corresponding components and sums the result. It is directly related to the cosine of the angle between the two vectors. When the coordinate axes are translated, the difference vectors have the same dot product as before, ensuring that measures of angles and projections remain unchanged.

Norm (Magnitude) of a Vector

The norm of a vector quantifies its length or magnitude. Calculated as the square root of the sum of the squares of its components, the norm is invariant under translation because translation affects only the position of the vector’s tail without changing its length. This property is fundamental in maintaining distances and angles in a translated coordinate system.

Angle between Vectors

The angle between two vectors can be computed using the cosine formula, which involves the dot product and the product of the vectors' norms. Since both the dot product and the norms are invariant under translation of axes, the angle between the vectors remains constant. This invariance reflects the rigidity of Euclidean geometry under translations.

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