Question

Identify the following surfaces: (a) \(|\mathbf{r}|=k ;\) (b) \(\mathbf{r} \cdot \mathbf{u}=l ;\) (c) \(\mathbf{r} \cdot \mathbf{u}=m|\mathbf{r}|\) for \(-1 \leq m \leq+1\); (d) \(|\mathbf{r}-(\mathbf{r} \cdot \mathbf{u}) \mathbf{u}|=n\) Here \(k\). I. \(m\) and \(n\) are fixed scalars and \(u\) is a fixed unit yector.

Short answer

(a) Sphere; (b) Plane; (c) Double-napped cone; (d) Cylinder.

Step-by-step solution

Understanding part (a)

The equation \(|\mathbf{r}| = k\) represents the set of all points \(|\mathbf{r}|\) that have a distance \(k\) from the origin. This describes a sphere centered at the origin with radius \(k\).

Understanding part (b)

The equation \(\mathbf{r} \cdot \mathbf{u} = l\) represents the set of all points \(\mathbf{r}\) whose dot product with a fixed unit vector \(\mathbf{u}\) is \(l\). This describes a plane perpendicular to \(\mathbf{u}\) and at a distance \(l\) from the origin.

Understanding part (c)

The equation \(\mathbf{r} \cdot \mathbf{u} = m|\mathbf{r}|\) where \(-1 \leq m \leq +1 \,\), represents points where the dot product of \(\mathbf{r}\) and \(\mathbf{u}\) is a scaled version of the magnitude of \(\mathbf{r}\). This describes a double-napped cone with \(\mathbf{u}\) as the axis, opening at an angle depending on \(m\).

Understanding part (d)

The equation \(|\mathbf{r}-(\mathbf{r} \cdot \mathbf{u}) \mathbf{u}| = n\) represents points \(\mathbf{r}\) which have a constant distance \(n\) from the line spanned by \(\mathbf{u}\). This describes a cylinder with radius \(n\) and axis along the vector \(\mathbf{u}\).

Key concepts

sphere equation

Understanding the equation of a sphere is fundamental in vector calculus. A sphere can be described with the equation \(|\mathbf{r}| = k\). Here, \(|\mathbf{r}|\) stands for the magnitude of vector \(\mathbf{r}\), which is the distance from the origin to point \(\mathbf{r}\). This equation tells us that every point \(\mathbf{r}\) on the sphere is exactly distance \(k\) from the origin.
In other words, think of a sphere as a perfect ball with all its surface points equidistant from a central point (the origin). The value \(k\) is called the radius of the sphere. For example, if \(k = 5\), the sphere has a radius of 5 units.
In a more general form, for a sphere not centered at the origin, the equation is: \((x - a)^2 + (y - b)^2 + (z - c)^2 = k^2\), where \((a, b, c)\) is the center and \(k\) is the radius.

plane equation

Next, we come to the equation of a plane. A plane in vector terms can be represented by \(\mathbf{r} \cdot \mathbf{u} = l\). Here, \(\mathbf{r}\) is any point on the plane, \(\mathbf{u}\) is a fixed unit vector perpendicular to the plane (normal vector), and \(l\) is the distance from the origin to the plane.
This equation tells us that every point \(\mathbf{r}\) on the plane creates a dot product with \(\mathbf{u}\) equal to \(l\). A dot product being equal to a constant means that the angle between \(\mathbf{r}\) and \(\mathbf{u}\) is maintained but varies in magnitude, keeping them at a consistent distance.
For example, if \(\mathbf{u} = (1, 0, 0)\) and \(l = 3\), the plane equation simplifies to \(x = 3\), which is a vertical plane 3 units away from the yz-plane.

cone equation

A cone in vector calculus can be tricky but fascinating! The equation \(\mathbf{r} \cdot \mathbf{u} = m|\mathbf{r}|\) (with \(- 1 \leq m \leq +1\)) represents a double-napped cone.
In simpler terms, this equation states that the dot product of vector \(\mathbf{r}\) and unit vector \(\mathbf{u}\) is a scaled version of the magnitude of \(\mathbf{r}\). The scalar \(m\) dictates the cone’s opening angle and direction.
Imagine \(\mathbf{u}\) as the central axis of the cone. The value of \(m\) determines the slant angle of the cone from this axis. When \(m = 0\), the cone opens symmetrically around \(\mathbf{u}\). When \(m = \pm 1\), the cone's sides are infinitely spread, resembling a plane.

cylinder equation

Finally, let's discuss the equation of a cylinder. The equation for a cylinder is given by \(|\mathbf{r} - (\mathbf{r} \cdot \mathbf{u}) \mathbf{u}| = n\), where \(n\) is the radius. Here, \(\mathbf{r}\) is any point on the cylinder, \(\mathbf{u}\) is the axis vector, and \(n\) is a fixed distance from the axis.
To break it down: \(\mathbf{r} \cdot \mathbf{u}\) calculates the projection of \(\mathbf{r}\) onto \(\mathbf{u}\). Subtracting this from \(\mathbf{r}\) gives a vector orthogonal to \(\mathbf{u}\). The magnitude of this vector is constantly \(n\), meaning every point \(\mathbf{r}\) is consistently \(n\) units away from the central axis, forming a cylinder.
For example, if \(\mathbf{u}=(0,1,0)\) and \(n=2\), the cylinder has a vertical axis along the y-axis and a radius of 2 units, like a soup can standing upright.