Question

Determine whether the two lines intersect. and if so, find the point of intersection. $$ \begin{array}{lll} x=1+2 t, & y=1-4 t, & z=5-t \\ x=4-v, & y=-1+6 v, & z=4+v \end{array} $$

Name: Problem 11, Chapter 14: Calculus
Uploaded: 2022-03-17T22:38:57-08:00
Duration: 3 min 31 s
Channel: Steven Clarke
Description: Determine whether the two lines intersect. and if so, find the point of intersection. x=1+2 t, y=1-4 t, z=5-t x=4-v, y=-1+6 v, z=4+v

   Determine whether the two lines intersect. and if so, find the point of intersection.
$$
\begin{array}{lll}
x=1+2 t, & y=1-4 t, & z=5-t \\
x=4-v, & y=-1+6 v, & z=4+v
\end{array}
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 14, Problem 11

Instant Answer

Solved by Expert Steven Clarke

03/17/2022

Step 1

If the lines intersect, the x and y components at the point of intersection must be equal. This gives us two equations: For x-components: \[1+2t = 4-v \tag{1}\] For y-components: \[1-4t = -1+6v \tag{2}\] Show more…

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Determine whether the two lines intersect. and if so, find the point of intersection. $$ \begin{array}{lll} x=1+2 t, & y=1-4 t, & z=5-t \\ x=4-v, & y=-1+6 v, & z=4+v \end{array} $$

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Key Concepts

Parametric Equations

Parametric equations describe a line by expressing its coordinates as functions of one or more parameters. In this representation, a line is described by a point through which it passes and a direction vector that indicates its orientation in space.

Intersection of Lines in 3D Space

Determining the intersection of lines in three-dimensional space involves checking if there exist parameter values that satisfy the equations of both lines simultaneously. If such values exist, substituting them back into the parametric equations gives the common point of intersection.

Systems of Linear Equations

When equating the parametric expressions of two lines, a set of linear equations is formed. Solving this system of equations allows us to find the parameter values that may indicate a point of intersection, and also verifies whether the system is consistent—i.e., whether the lines actually intersect.

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