Problem 31 - Advanced Engineering Mathematics

Question

In Problems, determine whether the given lines intersect. If so, find the point of intersection. $$ \begin{aligned} &x=4+t, y=5+t, z=-1+2 t \\ &x=6+2 s, y=11+4 s, z=-3+s \end{aligned} $$

Step-by-Step Solution

Answer

The lines intersect at the point \((2, 3, -5)\).

Step 1: Set Equations Equal

To find the point of intersection, set the equations for each coordinate equal to each other. This means setting the x-coordinates, y-coordinates, and z-coordinates of the two lines equal:\[4 + t = 6 + 2s\]\[5 + t = 11 + 4s\]\[-1 + 2t = -3 + s\]

Step 2: Solve the System of Equations

Solve the equations obtained from Step 1 simultaneously to find the values of \(t\) and \(s\):Start with the first equation:\[t = 6 + 2s - 4 \rightarrow t = 2 + 2s\]Substitute \(t\) from above into the second equation:\[5 + (2 + 2s) = 11 + 4s\]Simplify:\[7 + 2s = 11 + 4s\]\[-4 = 2s\]\[s = -2\]Substitute \(s = -2\) back to find \(t\):\[t = 2 + 2(-2) = 2 - 4 = -2\]

Step 3: Check for Consistency in the Third Equation

Verify that the values \(t = -2\) and \(s = -2\) satisfy the third equation:\[-1 + 2(-2) = -1 - 4 = -5\]\[-3 + (-2) = -3 - 2 = -5\]The left and right sides are equal, confirming the solution is consistent.

Step 4: Find the Intersection Point

Substitute \(t = -2\) into the parametric equations of the first line to find the point of intersection:\[x = 4 + (-2) = 2\]\[y = 5 + (-2) = 3\]\[z = -1 + 2(-2) = -1 - 4 = -5\]

Step 5: Conclusion

The lines intersect at the point \((2, 3, -5)\) as the parameters \(t = -2\) and \(s = -2\) satisfy the equations for each coordinate in both lines.

Key Concepts

Parametric Equations

Parametric equations are a way of expressing a set of values, typically the coordinates of points in a space, using one or more parameters. Instead of expressing the coordinates as explicit functions of each other, they are expressed as functions of a common parameter, usually denoted as "t" for time or another variable. This makes it easier to describe curves and lines, especially in multi-dimensional spaces like 2D and 3D.In the given problem, we have two lines represented by parametric equations:- First line: - \( x = 4 + t \) - \( y = 5 + t \) - \( z = -1 + 2t \)- Second line: - \( x = 6 + 2s \) - \( y = 11 + 4s \) - \( z = -3 + s \)Here, "t" and "s" are parameters representing any real number, which help generate points along each line. The point of intersection, if it exists, will occur where the parametric expressions for both lines provide the same coordinates.

System of Equations

To determine whether two lines intersect, we solve a system of equations derived from setting the parametric equations equal to each other. This involves finding a common solution for the parameters that satisfies all equations simultaneously. Let's break it down:- Set equal the parametric equations for each of the coordinates: * \( 4 + t = 6 + 2s \) * \( 5 + t = 11 + 4s \) * \( -1 + 2t = -3 + s \)- Solve these simultaneous equations by substitution or elimination: * From the first equation, isolate \( t \): - \( t = 6 + 2s - 4 = 2 + 2s \) * Substitute into the second equation: - \( 5 + t = 11 + 4s \) - \( 7 + 2s = 11 + 4s \) * Solve for \( s \): - \( -4 = 2s \) - \( s = -2 \) * Substitute \( s = -2 \) to find \( t \): - \( t = 2 + 2(-2) = -2 \)The solution \( t = -2 \) and \( s = -2 \) satisfies all conditions, indicating an intersection if verified by the third equation.

3D Coordinate Geometry

3D Coordinate Geometry is a method to study geometric figures in three dimensions using coordinate systems. In this context, we examine the properties and relations of lines, planes, and shapes within the spatial dimension.For lines in 3D space, each position along a line can be represented using parametric equations, while intersecting lines share a common point where their coordinates meet. The exercise requires us to verify whether lines described by parametric equations indeed meet by solving a system of equations.In the solution:- We confirm that the solution \( t = -2 \) and \( s = -2 \) provides the same coordinates in both lines, hence ensuring they do intersect at the point determined by: - \( x = 2 \) - \( y = 3 \) - \( z = -5 \)This intersection point \( (2, 3, -5) \) in 3D space illustrates the principle of 3D geometry where geometric figures’ spatial relations can be theoretically and practically calculated using algebraic expressions.