   Chapter 11.5, Problem 93E

Chapter
Section
Textbook Problem

# Finding the Distance Between Two Parallel Lines In Exercises 99 and 100, verify that the two lines are parallel and find the distance between the lines. L 1 : x = 2 − t , y = 3 + 2 t , z = 4 + t L 2 : x = 3 t , y = 1 − 6 t , z = 4 − 3 t

To determine
Find if the lines L1 and L2 are parallel and distance between them.

Explanation

Given:

The parametric equations of the line L1 are,

x=2t,y=3+2t and z=4+t

The parametric equations of the line L2 are,

x=3t,y=16t and z=43t

Explanation:

As we know the distance between a point and a line is,

D=PQ×uu

u — Direction vector for the line

P and Q — points on the lines respectively.

Now further two lines are parallel if their parallel vectors or direction vectors are proportional, which means,

v=ku

Here,

u and v — direction vectors of line L1 and L2 respectively

and k —any real number.

because the direction numbers of a line are the coefficients of parameter t in the parametric equation of the line.

Therefore,

The parametric equations of the line L1 are,

x=2t,y=3+2t and z=4+t

The parametric equations of the line L2 are,

x=3t,y=16t and z=43t

Now since, the direction numbers of a line are the coefficients of parameter t in the parametric equation of the line.

So, the direction vector of L1 is,

u=1,2,1

And, the direction vector of L2 is,

v=3,6,3=31,2,1

Also now, find the value of k then,

v=ku31,2,1=k1,2,1k=3

Also now, from the above calculation the value of k is 3.

Therefore, the provided lines are parallel.

Now let us find a point on the line L2

Substituting t=0, then obtain the point as,

For the x coordinate of the point,

x=3t=3(0)=0

For the y coordinate of the point,

y=16t=16(0)=1

For the z coordinate of the point,

z=43t=43(0)=4

From the above calculation the point is,

Q=(0,1,4).

Similarly, find a point on the line L1.

For the x coordinate of the point,

x=2t=20=2

For the y coordinate of the point,

y=3+2t=3+2(0)=3

For the z coordinate of the point,

z=43t=4+3(0)=4

From the above calculation the point is,

P=(2,3,4)

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