Chapter 12.5, Problem 78E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Find the distance between the skew lines with parametric equations *x* = 1 + *t*, *y* = 1 + 6*t*, *z* = 2*t*, and *x* = 1 + 2*s*, *y* = 5 + 15*s*, *z* = −2 + 6*s*.

To determine

**To find:** The distance between the skew lines with the parametric equations

Explanation

**Formula used:**

The expression to find the distance between two skew lines is,

Here,

**Calculation:**

The distance between two skew lines is the absolute value of the scalar projection of the vector that is joined between the points on the skew lines along the normal vector of two lines.

Write the parametric equation of the line

Rewrite the expression as follows.

Write the parametric equation of the line

Rewrite the expression as follows.

The expressions for the parametric equations for a line through the point

Compare equation (2) with equation (4) and write the direction vector of line

Compare equation (3) with equation (4) and write the direction vector of line

The direction vectors of two skew lines are written as follows.

The normal vector is the cross product of direction vectors of both the lines.

Write the expression to find normal vector as follows.

Substitute 1 for

Consider the scalar parameters *t* and *s* as 0 and obtain the two points on the two skew lines as follows.

Substitute 0 for *t* in equation (2) and obtain a point on the line

The point on the line

Substitute 0 for *s* in equation (3) and obtain a point on the line

The point on the line

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