Question

Relationship between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}.\) Consider the helix \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle,\) for \(-\infty

Short answer

The point on the helix at which \(r(t)\) and its tangent vector are orthogonal is \((1, 0, 0)\).

Step-by-step solution

Compute the tangent vector \(\mathbf{r}^{\prime}(t)\)

To determine the derivative of the helix function, differentiate each component of \(\mathbf{r}(t)\) with respect to t, which will give us the tangent vector: \(\mathbf{r}^{\prime}(t) = \Big\langle\frac{d}{dt}(\cos t), \frac{d}{dt}(\sin t), \frac{d}{dt}(t)\Big\rangle = \langle-\sin t, \cos t, 1\rangle\)

Determine where \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t)\) are orthogonal using the dot product

To determine the points at which \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t)\) are orthogonal, we need to find the points where their dot product is equal to zero: \(\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t) = 0\) Calculate the dot product: \((\cos t, \sin t, t) \cdot (-\sin t, \cos t, 1) = -\sin t \cos t + \sin t \cos t + t = 0\)

Find the corresponding t values at which this occurs

Now, we need to solve the equation we got in Step 2 to find the t values at which \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t)\) are orthogonal: \(t = 0\) So, there is just one t value for which \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) are orthogonal: \(t=0\).

Plug these t values into \(\mathbf{r}(t)\) to find the coordinates of the points

Finally, we need to plug the t value into \(\mathbf{r}(t)\) to find the coordinates of the point at which it is orthogonal: \(\mathbf{r}(0) = \langle\cos(0), \sin(0), 0\rangle = \langle1, 0, 0\rangle\) Thus, the point on the helix at which \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) are orthogonal is \((1, 0, 0)\).

Key concepts

Dot Product

The dot product, also known as the scalar product, is a significant operation in vector calculus that reflects the relationship between two vectors. It is essentially a way to multiply them, resulting in a scalar (a single number), rather than another vector. This operation is useful in determining the angle between the vectors, checking for orthogonality, or finding the projection of one vector on another.

The dot product is calculated by multiplying the corresponding entries of the vectors and summing these products:

For vectors \(\mathbf{a} = \langle a_1, a_2, ... , a_n \rangle\) and \(\mathbf{b} = \langle b_1, b_2, ..., b_n \rangle\),the dot product is \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + ... + a_nb_n\).

If the dot product is zero, the vectors are orthogonal, meaning they meet at a right angle. This property was crucial to solve the exercise, where the dot product was set to zero to find when the position vector and its derivative are orthogonal on the helix.

Tangent Vector

In vector calculus, a tangent vector represents the rate of change of a curve at a particular point and has the direction of the curve's tangent line at that point. When considering a path described by a vector-valued function, the tangent vector can be found by taking the derivative of that function with respect to its parameter.

For the vector function \(\mathbf{r}(t)\), the tangent vector is \(\mathbf{r}^{\prime}(t)\), which provides a vectorial expression for the trajectory's direction and speed. In the provided exercise, the tangent vector \(\mathbf{r}^{\prime}(t)\) was computed and used to determine where it is orthogonal to the position vector \(\mathbf{r}(t)\). This was an essential step in solving the problem and showcasing the concept's practical application.

Vector Calculus

Vector calculus is an extension of calculus to vector fields, where operations such as differentiation and integration are applied to vectors. It allows the analysis of vector functions and fields in multi-dimensional space and is fundamental in physics and engineering for describing and understanding force fields, velocity fields, and many other vector quantities.

In our exercise, vector calculus principles were employed to differentiate the helix's vector function and find an orthogonal relationship. Key operations include finding the derivative of vector functions, taking the dot product, and understanding the geometric implications of these procedures on space curves, such as helices.

Helix

A helix is a type of smooth space curve with a constant radius around a central axis. A common example of a helix is a spring or a corkscrew. Mathematically, a helix can be described by a vector function with trigonometric components to define its circular motion and another component to control its progression along the axis.

For instance, the function \(\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle\) represents a helix. In the context of the exercise, understanding the geometry of the helix was important in determining the points on this three-dimensional curve where the position vector and tangent vector are orthogonal. The complexity of the helix shape in the exercise adds a rich situation for the application of the aforementioned vector calculus concepts.