Mathematics

Problem

67. Distance from a Point to a Line Consider the line

L

with equation

2 x+5 y=10

and the point

P=(3,7)

. (a) Verify that

A=(0,2)

and

B=(5,0)

are the

y

-and

x

-intercepts of

L

. (b) Find

\mathbf{w}_{1}=\operatorname{proj}_{\overrightarrow{A B}} \overrightarrow{A P}

and

\mathbf{w}_{2}=\overrightarrow{A P}-\operatorname{proj}_{\overrightarrow{A B}} \overrightarrow{A P}

. (c) Writing to Learn Explain why

\left|\mathbf{w}_{2}\right|

is the distance from

P

L

. What is this distance? (d) Find a formula for the distance of any point

P=\left(x_{0}, y_{0}\right)

L

. (e) Find a formula for the distance of any point

P=\left(x_{0}, y_{0}\right)

to the line

a x+b y=c