Problem 11 - Analytical Geometry: 2D and 3D

Question

Find the equation of the right circular cylinder of radius 1 with axis as \(\frac{x-1}{2}=\frac{y}{3}=\frac{z-3}{1}\)

Step-by-Step Solution

Answer

Based on the information given and the step by step solution, the equation of the right circular cylinder of radius 1 with axis as \(\frac{x-1}{2}=\frac{y}{3}=\frac{z-3}{1}\) is: \(|(3y - 9)(y - 3)|^2 + (x - 2)^2 = 1\).

Step 1: Convert the symmetric equation to the parametric form

The symmetric equation of the axis is given by: \(\frac{x-1}{2}=\frac{y}{3}=\frac{z-3}{1}\) Let's introduce a parameter \(t\): \(x = 1 + 2t\) \(y = 3t\) \(z = 3 + t\) Which gives us the parametric equation of the axis: \(P(t) = (1 + 2t, 3t, 3 + t)\)

Step 2: Find a point on the axis

Evaluate \(P(t)\) at any value of \(t\) to get a point on the axis. Let's use \(t = 0\): \(P(0) = (1, 0, 3)\) So, \((1, 0, 3)\) is a point on the axis.

Step 3: Write the general equation of the cylinder

Let the axis direction be given by the vector \(\vec{d} = (2, 3, 1)\). Let any point on the cylinder be \(Q = (x, y, z)\). The cross product of the vector connecting the points on the axis and the cylinder \(\vec{PQ} = (x - 1, y, z - 3)\) and the axis direction vector \(\vec{d}\) is orthogonal to the axis. So, we have \(\vec{PQ} \times \vec{d} = \vec{M}\) \(|(x - 1) - 3(z - 3)|^2 + |-2(x - 1) + (z - 3)|^2 = R^2\) Now, we compute the cross product and simplify. Substitute \(R = 1\) and square both sides: \(|^3y - 9(y - 3)|^2 + |-(x - 1) - 1|^2 = 1\) \(|(3y - 9)(y - 3)|^2 + (x - 2)^2 = 1\) This is the equation of the right circular cylinder of radius 1 with axis as \(\frac{x-1}{2}=\frac{y}{3}=\frac{z-3}{1}\).

Key Concepts

Parametric Form

The parametric form of an equation is useful because it describes a path or collection of points using parameters. In the given problem, the axis of the cylinder is given in symmetric form as \( \frac{x-1}{2} = \frac{y}{3} = \frac{z-3}{1} \). By introducing a parameter \( t \), we can express each variable in terms of \( t \):

• \( x = 1 + 2t \)
• \( y = 3t \)
• \( z = 3 + t \)

This leads to the parametric form \( P(t) = (1 + 2t, 3t, 3 + t) \). Parametric equations simplify the representation of lines or curves, allowing us to visualize and compute properties like intersections and directions more easily. It's like giving each point a unique time-stamp based on \( t \). This conversion makes further calculations, such as finding specific points or directions, much more straightforward.

Cylinder Axis

Understanding the cylinder axis is crucial for defining the properties and equations of a cylinder. The axis of a cylinder is a straight line about which the cylinder is symmetric and is defined by the parametric equation \( P(t) = (1 + 2t, 3t, 3 + t) \). In this geometry problem, the direction of the axis is crucial and is represented by a vector \( \vec{d} = (2,3,1) \). This axis acts as a spine,

• Defining the orientation of the cylinder.
• Allowing calculations and relationships involving the radius and surface.
• Providing a reference for creating symmetrical parts of the cylinder around it.

Learning to extract the axis correctly from a given symmetric equation is key to analyzing cylinders in three dimensions effectively.

Cross Product

The cross product is an essential operation in vector mathematics, allowing us to find a vector orthogonal to two given vectors. When determining the equation for a right circular cylinder, the cross product helps identify vectors perpendicular to the cylinder's axis. Here, we consider the vector connecting the cylinder's surface and a point on the axis, denoted as \( \vec{PQ} = (x - 1, y, z - 3) \).The axis direction vector is \( \vec{d} = (2, 3, 1) \). Using the cross product, \( \vec{PQ} \times \vec{d} \), we find a vector orthogonal to both, which helps correctly express the cylinder's geometry.

• This vector must satisfy certain conditions based on the radius given.
• It encapsulates the cylindrical symmetry and our directional understanding of it.
• Squares of its components are used to frame the cylinder's algebraic equation.

Utilizing the cross product in this manner ensures a full grasp of the spatial orientation and boundaries of the cylinder.

Symmetric Equation

The symmetric equation serves as a starting point for understanding the orientation and direction of lines in three-dimensional space. The given symmetric equation, \( \frac{x-1}{2} = \frac{y}{3} = \frac{z-3}{1} \), offers a concise way to express the condition where multiple variables relate through certain ratios. Its advantage lies in simplicity and clarity, capturing the equality of proportions without extensive arithmetic.

• It allows conversion to parametric equations easily for further calculations.
• The symmetric form is particularly handy for identifying axis directions directly.
• Ratios reflect constant speeds or flows across dimensions.

Understanding this form helps transition from abstract relationships to more tangible parametric representations, which is invaluable in more complex geometric analyses involving axis-aligned features.