Question

The lines \(\mathrm{x}-3=[(\mathrm{y}+2) /(-1)]=\mathrm{z}-\mathrm{I}\) and \((\mathrm{x} / 2)=[(\mathrm{z}+3) / 3], \mathrm{y}+\mathrm{I}=0\) intersects at (A) \((2,1,0)\) (B) \(\overline{(-2,1}, 0)\) (C) \((-2,-1,0)\) (D) \((2,-1,0)\)

Short answer

The intersection point of the given lines is \((2, 5, 0)\), but this point is not included in the given options. There might be an error in the given problem or options.

Step-by-step solution

Find the parametric equations of the lines

Given lines are: 1. \(\mathrm{x}-3=[(\mathrm{y}+2) /(-1)]=\mathrm{z}-\mathrm{I}\) 2. \((\mathrm{x} / 2)=[(\mathrm{z}+3) / 3], \mathrm{y}+\mathrm{I}=0\) Let's rewrite them as parametric equations: 1. Line 1: \[\begin{cases} x = 3 - t \\ y = -2 - t \\ z = I + t \end{cases}\] 2. Line 2: \[\begin{cases} x = 2s \\ y = - I \\ z = -3 + 3s \end{cases}\]

Find the intersection point

To find the intersection point, we will equate the corresponding x, y, and z components of both lines' parametric equations: \[\begin{cases} 3 - t = 2s \\ -2 - t = - I \\ I + t = -3 + 3s \end{cases}\] Solve this system of equations to find the values of \(t\) and \(s\): From equation 1: \(t = 3 - 2s\) Substitute the value of \(t\) in equation 2 and equation 3: \[\begin{cases} -2 - (3 - 2s) = - I \\ I + (3 - 2s) = -3 + 3s \end{cases}\] On simplifying the equations, we get: \[\begin{cases} 2s -H = - I \\ -H +5 - 2s = -3 + 3s \end{cases}\] By solving these two equations, we get: \(s = 1\) and \(H = -5\) Now plug the value of \(s\) back into Line 2's parametric equations to find the intersection point: \[\begin{cases} x = 2(1) \\ y = - (-5) \\ z = -3 + 3(1) \end{cases}\] \[\begin{cases} x = 2 \\ y = 5 \\ z = 0 \end{cases}\] The intersection point is \((2, 5, 0)\). However, none of the given options match this point. So, there might be an error in the given problem or options.

Key concepts

Parametric Equations

Parametric equations are a way to express a mathematical relationship by using parameters. In the context of 3D coordinate geometry, parametric equations are extremely useful as they allow us to represent the position of a point on a line as a function of a single parameter.

Consider a line in 3D space. We can create a set of parametric equations that uniquely describe any point on this line by considering an arbitrary parameter, say 't' for line 1 and 's' for line 2 in our exercise. Each coordinate (x, y, z) of the line can be expressed as a separate equation involving 't' or 's'. The advantage is that by changing the value of 't' or 's', we can precisely traverse along the entire line.

For example, the first line in the exercise is represented by the set of equations where 't' is our parameter. These provide a clear path to plot the line and understand its direction in the 3D space. It's imperative when trying to find where two lines intersect, as each point on both lines will correspond to particular values of their respective parameters.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. When solving problems involving the intersection of lines in 3D space, it is often necessary to create such a system to find the point, if any, at which the lines cross.

In the case of the exercise provided, we can create a system of equations using the parametric equations of each line. By equating the x, y, and z components from each line, we're looking for a common solution that satisfies all equations simultaneously – this solution will correspond to the intersection point.

Steps to Solve a System

• Write Out the Equations: First, we express each line with its respective parameter.
• Combine Equations: Next, we set the parametric equations from both lines equal to each other along each dimension (x, y, and z).
• Solve The System: We manipulate the equations algebraically to solve for the parameters. This generally involves substitution or elimination methods.

Learning to solve these systems is critical, as it extends to finding solutions in various disciplines such as physics, engineering, and economics.

3D Coordinate Geometry

3D coordinate geometry deals with points, lines, and planes in three-dimensional space. We use three axes, typically labeled x, y, and z, where each point in space is defined by a set of three coordinates.

The understanding of 3D geometry is crucial when trying to conceptualize the intersection of lines in higher dimensions. It allows us to extend the familiar principles of shapes, areas, and distances from 2D plane geometry to 3D space.

Applications in 3D Space

This form of geometry is not just a theoretical construct; it has practical applications in various fields such as computer graphics for rendering 3D models, architecture for designing buildings and spaces, and in physics for analyzing the trajectories of objects through space.

An intuitive grasp of 3D coordinate geometry is necessary when working with parametric equations and solving systems of equations, as seen in the given exercise. By mastering these coordinates, students can visualize mathematical problems more concretely and solve complex spatial issues with confidence.