Question

Find \(\|\vec{r}(t)\|\). $$ \vec{r}(t)=\left\langle t, t^{2}\right\rangle $$

Short answer

\( \|\vec{r}(t)\| = |t| \cdot \sqrt{1 + t^2} \).

Step-by-step solution

Understand the Vector Notation

The vector function \( \vec{r}(t) = \langle t, t^2 \rangle \) represents a vector in two dimensions, where the first component is \( t \) and the second component is \( t^2 \). Each point on this vector's path can be represented as \( (t, t^2) \).

Recall the Norm Formula for 2D Vectors

The norm (or magnitude) of a vector \( \langle a, b \rangle \) in two dimensions is calculated using the formula: \[ \| \langle a, b \rangle \| = \sqrt{a^2 + b^2} \] We will apply this formula to \( \vec{r}(t) = \langle t, t^2 \rangle \) to find its magnitude.

Identify Components

Identify the components of the vector function specific to \( \vec{r}(t) \). For this vector: \( a = t \) and \( b = t^2 \).

Apply the Norm Formula

Substitute \( a = t \) and \( b = t^2 \) into the magnitude formula: \[ \| \vec{r}(t) \| = \sqrt{t^2 + (t^2)^2} \] This simplifies to:\[ \| \vec{r}(t) \| = \sqrt{t^2 + t^4} \]

Simplify the Expression

Combine like terms in the square root: \[ \| \vec{r}(t) \| = \sqrt{t^2(1 + t^2)} \] This can be written as: \[ \| \vec{r}(t) \| = |t| \cdot \sqrt{1 + t^2} \] Note: The absolute value is used because the magnitude must be non-negative.

Key concepts

Vector Function

A vector function is simply a function where the inputs (like time or any parameter) yield vectors as outputs. It is a crucial concept in vector calculus as it allows tracking the movement or state of an object in a vector space over time. In the example \( \vec{r}(t) = \langle t, t^2 \rangle \), each input value of \( t \) produces a specific vector \( \langle t, t^2 \rangle \).
The output vector provides us two pieces of information:

• The first component (\( t \)) often represents a direction on the x-axis.
• The second component (\( t^2 \)) typically corresponds to the y-axis direction.

Understanding this, you can trace the path of the vector in a two-dimensional plane, plotting points like \( (t, t^2) \) as \( t \) changes. As \( t \) varies, you can visualize how the vector traverses a parabola.

Magnitude of a Vector

The magnitude of a vector is a measure of its length. Think of it as the size or the norm of the vector. For a vector expressed as \( \langle a, b \rangle \) in two-dimensional space, the magnitude can be found using the formula:\[\| \langle a, b \rangle \| = \sqrt{a^2 + b^2}\]
In the context of our exercise with \( \vec{r}(t) = \langle t, t^2 \rangle \), you're asked to find the magnitude of vectors at different points in time, \( t \). You substitute \( a = t \) and \( b = t^2 \) into the magnitude formula. This leads to:\[\| \vec{r}(t) \| = \sqrt{t^2 + (t^2)^2}\]
Upon simplifying, the expression becomes:\[\| \vec{r}(t) \| = \sqrt{t^2(1 + t^2)} = |t| \cdot \sqrt{1 + t^2}\] The absolute value \( |t| \) ensures the magnitude is non-negative, aligning with the vector's size perception.

Two-Dimensional Vectors

Two-dimensional vectors are vectors with two components, each representing a different axis in a coordinate system, usually rendered as \( \langle x, y \rangle \). These vectors allow us to describe two-plane motion fully, necessary in physics, engineering, and computer graphics.
In two-dimensional vectors, both position and direction can be captured intuitively:

• The horizontal component (\( x \)) often aligns with the x-axis.
• The vertical component (\( y \)) corresponds with the y-axis.

The understanding of these vectors is crucial as it lays the groundwork for more advanced topics like vector calculus and 3D vector graphics. In the provided example, the vector \( \vec{r}(t) = \langle t, t^2 \rangle \) has its direction and magnitude easily visualized on the plane. This direct exploration of positions makes two-dimensional vectors a fundamental concept in understanding various scientific and practical phenomena.