Formulas (1), (2), (3), (5), and (10), which apply to planes in 3-space, have analogs for lines in 2-space. (a) Draw an analog of Figure 11.6.3 in 2-space to illustrate that the equation of the line that passes through the point P x 0 , y 0 and is perpendicular to the vector n = a , b can be expressed as n ⋅ r − r 0 = 0 where r = x , y and r 0 = x 0 , y 0 . (b) Show that the vector equation in part (a) can be expressed as a x − x 0 + b y − y 0 = 0 This is called the point-normal form of a line. (c) using the prof of Theorem 11.6.1 as a guide, show that is a and b are not both zero, then the graph of the equation a x + b y + c = 0 is a line that has n = a , b as a normal. (d) Using the proof of Theorem 11.6.2 as a guide, show that the distance D between a point P x 0 , y 0 and the line a x + b y + c = 0 is D = a x 0 + b y 0 + c a 2 + b 2 (e) Use the formula in part (d) to find the distance between the point P − 3 , 5 and the line y = − 2 x + 1.
Formulas (1), (2), (3), (5), and (10), which apply to planes in 3-space, have analogs for lines in 2-space. (a) Draw an analog of Figure 11.6.3 in 2-space to illustrate that the equation of the line that passes through the point P x 0 , y 0 and is perpendicular to the vector n = a , b can be expressed as n ⋅ r − r 0 = 0 where r = x , y and r 0 = x 0 , y 0 . (b) Show that the vector equation in part (a) can be expressed as a x − x 0 + b y − y 0 = 0 This is called the point-normal form of a line. (c) using the prof of Theorem 11.6.1 as a guide, show that is a and b are not both zero, then the graph of the equation a x + b y + c = 0 is a line that has n = a , b as a normal. (d) Using the proof of Theorem 11.6.2 as a guide, show that the distance D between a point P x 0 , y 0 and the line a x + b y + c = 0 is D = a x 0 + b y 0 + c a 2 + b 2 (e) Use the formula in part (d) to find the distance between the point P − 3 , 5 and the line y = − 2 x + 1.
Formulas (1), (2), (3), (5), and (10), which apply to planes in 3-space, have analogs for lines in 2-space.
(a) Draw an analog of Figure 11.6.3 in 2-space to illustrate that the equation of the line that passes through the point
P
x
0
,
y
0
and is perpendicular to the vector
n
=
a
,
b
can be expressed as
n
⋅
r
−
r
0
=
0
where
r
=
x
,
y
and
r
0
=
x
0
,
y
0
.
(b) Show that the vector equation in part (a) can be expressed as
a
x
−
x
0
+
b
y
−
y
0
=
0
This is called the point-normal form of a line.
(c) using the prof of Theorem 11.6.1 as a guide, show that is a and b are not both zero, then the graph of the equation
a
x
+
b
y
+
c
=
0
is a line that has
n
=
a
,
b
as a normal.
(d) Using the proof of Theorem 11.6.2 as a guide, show that the distance D between a point
P
x
0
,
y
0
and the line
a
x
+
b
y
+
c
=
0
is
D
=
a
x
0
+
b
y
0
+
c
a
2
+
b
2
(e) Use the formula in part (d) to find the distance between the point
P
−
3
,
5
and the line
y
=
−
2
x
+
1.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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