GRAPHS

• Graph 1 — Two parametrized curves in the \(xy\)-plane with position, tangent, and unit-tangent vectors.
• Graph 2 — A smooth curve.
• Graph 3 — A curve with a cusp point.
• Graph 4 — The parabola \(y = 3 - t^2\).
• Graph 5 — The point of intersection between a circle and a parabola, with tangent vectors.

EXERCISE 1 — Tangent Vectors & Tangent Lines

Background & Notes

The derivative \(\vec{r}'(t)\) of a vector function \(\vec{r}'(t)\) can be computed by differentiating the coordinate (aka parametric) functions.

In the \(xy\)-plane:

\[\begin{align*}
\vec{r}(t) &= \big< x(t), y(t) \big>\\
\vec{r}'(t) &= \big< x'(t), y'(t) \big>\\
\end{align*}\]

In 3-space:

\[\begin{align*}
\vec{r}(t) &= \big< x(t), y(t), z(t) \big>\\
\vec{r}'(t) &= \big< x'(t), y'(t), z'(t) \big>\\
\end{align*}\]

When \(\vec{r}'(t) \neq \vec{0}\):

• \(\vec{r}'(t)\) is tangent to the curve \(\vec{r}(t)\), and points in the direction of motion as \(t\) increases.
• \(\vec{r}'(t)\) defines the direction of the tangent line of \(\vec{r}(t)\).
• \(\vec{r}'(t)\) defines a unit tangent vector \(\hat{T}(t)\):

\[\hat{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}\]

Notation: In Leibniz notation, the vector derivative can be written \(\frac{d}{dt} \vec{r}(t) = \frac{d\vec{r}}{dt}\).

Assumptions

Let \(\vec{r}_1(t)\) and \(\vec{r}_2(t)\) be the plane curves:

\[\begin{align*}
\vec{r}_1(t) &= \big< t, t^2 \big>\\
\vec{r}_2(t) &= \big< t^3, t^2 \big>
\end{align*}\]

Instructions

1. Compute

Compute the tangent vectors \(\vec{r}_1'(t), \vec{r}_2'(t)\) and unit tangent vectors \(\hat{T}_1(t) = \frac{\vec{r}_1'(t)}{|\vec{r}_1'(t)|}, \hat{T}_2(t) = \frac{\vec{r}_2'(t)}{|\vec{r}_2'(t)|}\).

Identify any values of \(t\) where either \(\vec{r}_1(t) = \vec{0}\) or \(\vec{r}_2(t) = \vec{0}\).

2. Graph

Go to graph 1, and in the control window, identify the definitions for the two curves \(\vec{r}_1(t)\) (red) and \(\vec{r}_2(t)\) (green). Activate the curves using their checkboxes at the top-left.

Use the t-slider associated with each to curve to increase the value of \(t\) from \(t = -10\) to \(t = 10\), and observe the behavior of each curve’s position vector \(\vec{r}(t)\) (blue), tangent vector \(\vec{r}'(t)\) (black), and unit tangent vector \(\hat{T}(t)\) (purple).

Answer these questions:

• What is the relationship between the direction of the tangent and unit tangent vectors \(\vec{r}'(t), \hat{T}(t)\) and the curve?
• Describe the behavior of the second curve \(\vec{r}_2(t)\) at the point where \(\vec{r}_2'(t) = \vec{0}\).

3. Challenge — Equation of Tangent Line

If the vector derivative \(\vec{r}'(t) \neq \vec{0}\), \(\vec{r}'(t)\) defines the direction of the tangent line of \(\vec{r}(t)\), so the equation of the tangent line to \(\vec{r}(t)\) at time \(t\) is:

\[L(c) = c \vec{r}'(t) + \vec{r}(t)\]

For the curve \(\vec{r}_1(t) = \big< t, t^2 \big>\):

• Calculate the position and tangent vectors \(\vec{r}_1(t)\) and \(\vec{r}_1'(t)\) when \(t = 1\).
• Use the position and tangent vectors \(\vec{r}_1(1)\) and \(\vec{r}_1'(1)\) to derive the equation of the tangent line to \(\vec{r}_1(t)\) at the point \((1,1)\).
• Identify the coordinate functions \(x(c)\), \(y(c)\), \(z(c)\) for the tangent line, then go to graph 2 and add the tangent line to the graph, using the drop-down menu to select “space curve”. When you add the coordinate functions to the graph, you will need to change \(c\) to \(t\).

Do the same for the second curve \(\vec{r}_2(t) = \big< t^3, t^2 \big>\). Once you have the equation of the tangent line to \(\vec{r}_2(t)\) when \(t = 1\), add it to graph 3.

4. Double-Challenge — Tangent Lines

Find the time \(t\) when the tangent line to the parabola:

\[\vec{r}_1(t) = \big< t, t^2 \big>\]

is parallel to the line:

\[y = x/2\]

EXERCISE 2 — Tangent Vectors & Angles

Background & Notes

Since vector functions and their derivatives are vectors, you can define orthogonality and angles between them using the dot product.

Instructions

1. Compute — Orthogonality of \(\vec{r}(t)\) & \(\vec{r}'(t)\).

Graph 4 — Parametrize the graph \(y = 3 - t^2\) using the vector function:

\[\vec{r}(t) = \big< t, 3 - t^2 \big>\]

Find all values of \(t\) where the tangent vector \(\vec{r}'(t)\) is orthogonal to the position vector \(\vec{r}(t)\).

2. Compute — Angle of Intersection

If two curves intersect, their angle of intersection is the angle \(0 \leq \theta \leq \pi\) between their tangent vectors at the point of intersection.

Graph 5 — Parametrize the unit circle \(x^2 + y^2 = 1\) and the parabola \(y = x^2 - 1\) using the vector functions:

\[\begin{align*}
\vec{r}_C(t) = \big< \cos t, \sin t \big>\\
\vec{r}_P(s) = \big< s, s^2 - 1 \big>\\
\end{align*}\]

• Show that one of the points of intersection of the circle and the parabola has coordinates \((1,0)\), which occurs when \(t = 0\) and \(s = 1\) — that is, \(\vec{r}_C(0) = \vec{r}_P(1) = \big< 1, 0 \big>\).
• Compute and evaluate the tangent vectors \(\vec{r}_C(0)\) and \(\vec{r}_P(1)\).
• Use the tangent vectors to compute the angle of intersection \(\theta\) of this circle and parabola at the point \((1,0)\).

To compute the angle of intersection, either use this this code, or compute by hand using the dot product:

\[\theta = \arccos\left(\frac{\vec{r}_C(0) \cdot \vec{r}_P(1)}{|\vec{r}_C(0)||\vec{r}_P(1)|}\right)\]

BIG IDEAS

Tangent Vectors & Tangent Lines (exercise 1)

The derivative \(\vec{r}'(t)\) of a vector function \(\vec{r}'(t)\) can be computed by differentiating the coordinate (aka parametric) functions.

In the \(xy\)-plane:

\[\begin{align*}
\vec{r}(t) &= \big< x(t), y(t) \big>\\
\vec{r}'(t) &= \big< x'(t), y'(t) \big>\\
\end{align*}\]

In 3-space:

\[\begin{align*}
\vec{r}(t) &= \big< x(t), y(t), z(t) \big>\\
\vec{r}'(t) &= \big< x'(t), y'(t), z'(t) \big>\\
\end{align*}\]

When \(\vec{r}'(t) \neq \vec{0}\):

• \(\vec{r}'(t)\) is tangent to the curve \(\vec{r}(t)\), and points in the direction of motion as \(t\) increases.
• \(\vec{r}'(t)\) defines the direction of the tangent line of \(\vec{r}(t)\):

\[L(c) = c \vec{r}'(t) = \vec{r}(t)\]

• \(\vec{r}'(t)\) defines a unit tangent vector \(\hat{T}(t)\):

\[\hat{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}\]

Tangent Vectors & Angles (exercise 2)

Since vector functions and their derivatives are vectors, you can define orthogonality and angles between them using the dot product.

If two curves intersect, their angle of intersection is the angle \(0 \leq \theta \leq \pi\) between their tangent vectors at the point of intersection.