Consider the vector-valued function 𝐫(t)=t^2 𝐢+(t-3) 𝐣+t 𝐤 . Write a vector-valued function 𝐬(t)

step 1 Okay, we are given a vector valued function, and we've been asked to rewrite it with some of the

Consider the vector-valued function 𝐫(t)=t^2 𝐢+(t-3) 𝐣+t 𝐤 ....

Consider the vector-valued function $\mathbf{r}(t)=t^{2} \mathbf{i}+(t-3) \mathbf{j}+t \mathbf{k} .$ Write a vector-valued function $\mathbf{s}(t)$ that is the specified transformation of $\mathbf{r}$. (a) A vertical translation three units upward (b) A horizontal translation two units in the direction of the negative $x$ -axis (c) A horizontal translation five units in the direction of the positive $y$ -axis

Consider the vector-valued function
$\mathbf{r}(t)=t^{2} \mathbf{i}+(t-3) \mathbf{j}+t \mathbf{k} .$
Write a vector-valued function $\mathbf{s}(t)$ that is the specified transformation of $\mathbf{r}$.
(a) A vertical translation three units upward
(b) A horizontal translation two units in the direction of the negative $x$ -axis
(c) A horizontal translation five units in the direction of the positive $y$ -axis

Key Concepts

Vector-valued Functions

A vector-valued function assigns a vector to every real number (or parameter) t. It is typically expressed in component form, where each component is a function of t. This concept is fundamental in analyzing curves and motions in multi-dimensional space, allowing us to study trajectories, velocities, and accelerations of moving objects.

Transformations of Vector-valued Functions

Transformation of a vector-valued function often involves modifying the function's output by adding constants or by altering its input. These transformations can include translations, rotations, dilations, and reflections. They are used to shift or reshape the graph of the function without altering its essential properties like curvature.

Vertical Translation

A vertical translation shifts the entire graph of the function upward or downward by adding a constant to the component representing the vertical direction. In the context of a vector-valued function, this is typically accomplished by adding a constant vector whose nonzero entry corresponds to the vertical coordinate, resulting in a new function that is lifted or lowered in space.

Horizontal Translation

A horizontal translation moves the graph of the function left or right by altering the input variable inside the function’s expressions. This is accomplished by replacing t with (t - constant) in the appropriate component functions, effectively shifting the function along the horizontal axis without changing its shape.

Parameter Transformation

Parameter transformation involves substituting the variable t with an expression (like t - a constant) in order to achieve a desired shift in the graph. This concept is key when performing horizontal translations, as it adjusts the domain of the function while preserving the inherent relation between the components.

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