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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Show that the lines with symmetric equations x = y = z and x + 1 = y/2 = z/3 are skew, and find the distance between these lines.

To determine

To show: The lines with symmetric equations x=y=z and x+1=y2=z3 as skew lines.

Explanation

The two lines must be parallel, skew or intersect lines. If the two lines are parallel, the direction vectors of both the lines are scalar multiples of each other.

The two lines L1 and L2 are in the form of symmetric equations.

The symmetric equation for the line L1 is written as follows.

x=y=z

Rewrite the equations as follows.

x01=y01=z01 (1)

Write the expressions for the symmetric equations for a line through the point (x0,y0,z0) and parallel to the direction vector a,b,c.

xx0a=yy0b=zz0c (2)

Compare equations (1) and (2), and write the direction vector of line L1.

v1=1,1,1

The symmetric equation for the line L2 is written as follows.

x+1=y2=z3

Rewrite the expression as follows.

x(1)1=y02=z03 (3)

Compare equations (2) and (3), and write the direction vector of line L2.

v2=1,2,3

By observing the direction vectors of two lines L1 and L2, it is clear that the two lines are not scalar multiples of each other.

v1kv2

Therefore, the two line L1 and L2 are not parallel lines.

If the two lines have an intersection point, the parametric equations of the two lines must be equal.

Write the expressions for the parametric equations for a line through the point (x0,y0,z0) and parallel to the direction vector a,b,c.

x=x0+at,y=y0+bt,z=z0+ct (4)

From equation (1), it is clear that the point and the direction vector (v1) for line L1 are (0,0,0) and 1,1,1 respectively

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Chapter 12 Solutions

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