   Chapter 11.5, Problem 90E

Chapter
Section
Textbook Problem

# Finding the Distance Between a Point and a LineIn Exercises 95–98, find the distance between the point and the line given by the set of parametric equations. ( 1 , − 2 , 4 ) ;     x = 2 t , y = t − 3 , z = 2 t + 2

To determine

To calculate: For the given line and point, Find the distance between the given point and the given line x=2t,y=t3 and z=2t+2.

Explanation

Given:

The parametric equations of a line are:

x=2t,y=t3 and z=2t+2

And the provided point Q is (1,2,4).

Formula used:

The distance between a point Q and a line is:

D=PQ×uu

Here, D is the distance between point and the line, u is a direction vector for the line and P is a point on the line.

Calculation:

In the parametric equations of the line the direction numbers of a line are the coefficients of parameter t

Equations of the line are:

x=2t,y=t3 and z=2t+2

Direction vector is,

u=a,b,c=2,1,2

Substituting t=0, to find a point on a line, then obtain the point as:

For the x coordinate of the point:

x=2t=2(0)=0

For the y coordinate of the point,

y=t3=03=3

For the z coordinate of the point:

z=2t+2=2(0)+2=2

Now from the above calculation the point P on the line is:

P=(0,3,2)

Now, find the vector PQ

So, the vector PQ extending from point P to point Q will be

PQ=(10),(2(3)),(42)=1,1,2

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