   Chapter 11.5, Problem 94E

Chapter
Section
Textbook Problem

# Finding the Distance Between Two Parallel LinesIn Exercises 99 and 100, verify that the two lines are parallel and find the distance between the lines. L 1 : x = 3 + 6 t ,       y = − 2 + 9 t ,     z = 1 − 12 t L 2 : x = − 1 + 4 t ,     y = 3 + 6 t ,     z = − 8 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=3+6t,y=2+9t and z=112t

The parametric equations of the line L2 are,

x=1+4t,y=3+6t and z=8t

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=3+6t,y=2+9t and z=112t

The parametric equations of the line L2 are,

x=1+4t,y=3+6t and z=8t

So, the direction vector of L1 is,

u=6,9,12

And, the direction vector of L2 is,

v=4,6,8=236,9,12

Also now, find the value of k then,

v=ku236,9,12=k6,9,12k=23

Also now, from the above calculation the value of k is 23.

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=1+4t=1+4(0)=1

For the y coordinate of the point,

y=3+6t=3+6(0)=3

For the z coordinate of the point,

z=8t=8(0)=0

From the above calculation the point is,

Q=(1,3,0).

Similarly, find a point on the line L1.

For the x coordinate of the point,

x=3+6t=3+6(0)=3

For the y coordinate of the point,

y=2+9t=2+9(0)=2

For the z coordinate of the point,

z=112t=112(0)=1

From the above calculation the point is,

P=(3,2,1)

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