   Chapter 12.5, Problem 70E

Chapter
Section
Textbook Problem

# Use the formula in Exercise 12.4.45 to find the distance from the point to the given line.(0, 1, 3); x = 2t, y = 6 − 2t, z = 3 + t

To determine

To find: The distance from the point (0,1,3) to the line x=2t,y=62t,z=3+t .

Explanation

Consider the line L passes through the points Q and R .

Consider a point P(x0,y0,z0)=(0,1,3) which is not on the line L .

Formula:

Write the expression to find the distance from the point P(x0,y0,z0) to the line L .

d=|a×b||a| (1)

Here,

a is the direction vector from the point Q to the point R and

b is the direction vector from the point Q to the point P(x0,y0,z0) .

The points Q and R are determined by varying the scalar parameter t in the line equation.

Write the symmetric equations of the line as follows.

x=2t,y=62t,z=3+t (2)

Consider the value of scalar parameter t as 0 and obtain the point Q .

Substitute 0 for t in equation (2) to obtain the point Q .

x=2(0),y=62(0),z=3+0x=0,y=6,z=3

The point Q on the line is (0,6,3) .

Consider the value of scalar parameter t as 1 and obtain the point R .

Substitute 1 for t in equation (2) to obtain the point R .

x=2(1),y=62(1),z=3+1x=2,y=4,z=4

The point R on the line is (2,4,4) .

Write the expression to find direction vector from the point P(x1,y1,z1) to Q(x2,y2,z2) .

PQ=(x2x1),(y2y1),(z2z1) (3)

Calculation of direction vector a(QR) :

Substitute 0 for x1 , 6 for y1 , 3 for z1 , 2 for x2 , 4 for y2 , and 4 for z2 in equation (3),

a=(20),(46),(43)=2,2,1=2i2j+k

Calculation of direction vector b(QP) :

Substitute 0 for x1 , 6 for y1 , 3 for z1 , 0 for x2 , 1 for y2 , and 3 for z2 in equation (3),

b=(00),(16),(33)=0,5,0

Write the expression to find cross product of direction vectors

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### If p is a polynomial, Show that limxa p(x) = p(a)

Single Variable Calculus: Early Transcendentals, Volume I

#### Evaluate the iterated integral: 010x1x2dydx

Calculus: Early Transcendental Functions (MindTap Course List)

#### Multiply. (2cos+3)(4cos5)

Trigonometry (MindTap Course List)

#### The scalar projection of on is:

Study Guide for Stewart's Multivariable Calculus, 8th 