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for the plane through the point p = (.x0, yo, zu) and having the normal vector n =

(a, b, c), If x = (x, y, z) is an arbitrary point on the plane, then x - p will be a vector

in the plane and consequently will be perpendicular to n, as in Figure 10.30.

Recalling that two vectors are perpendicular if and only if their dot product is

zero, we write

0 = n (x - p) = (a, b, c) (x XC,, y - yo, z - z,,),

a(x ˜ I,,) + b@ - yr,) + c(z 20) = 0.

01 (17)

Form (17) is called the point-normal equation of the plane. It is sometimes

written as

ax + by + cz = d, (18)

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PLANES

Figure

Plum through p with normal n. 10.30

where, in this case, d = axe + byвЂќ + cro. Conversely, one can see that equation (18)

is the equation of the plane which has normal vector (a, b, c) and which contains

each of the points (90, d/c), (0, d/b, 0), and (d/a, 40).

Emmple 10.5 The equation of the plane through the point (1,2,3) with normal

˜

vector (4, 5. 6) is

4(r 1) + s(J 2) + 6(z - 3) = 0

4x + 5y + hz = 32.

or

Example 10.6 The equation 3x y + 42 = 12 is a nonparametric equation of

the plane through the pomt (4, 0. 0) (or (0, 0, 3) or (0, 12, 0) OT (5, 7, 1)) with

IвЂњвЂњ*ma1 xctor n = ( 3 >I 4>) .

To go from a nonparametric equation (18) of a plane to a parametric one, just

use (18) to find three points on the plane and then use equation (IS). It is more

difficult to go from a parametric representation to a nonparametric one, because

we need to find a normal n to the plane given vectors v and w parallel to the plane.

There arc two ways to compute such an n. First, one may use the exercises in the

last section and take n to be the cross product Y X w. Alternatively, given v and

w, une can solve the system of equations n. v = 0 and n w = 0 explicitly for n.

Example 10.7 To find the point-normal equation of the plane P which contains

the points

p = (2, 1, l), q = (1, 0, ˜3), and r = (0, I, 7),

230 EVCLIDEAN SPACES I1 01

note that vectors

vвЂќq-p=(pl,-l,p4) and u-r-p=(-2,0,6)

both lie on 2вЂ™. To find a normal n=(n,, nl. nz) to T, solve the system

-Ill - вЂњ2 4n3 =0

вЂњвЂ˜Y =

вЂќ вЂќ = -2n, + On2 + 6n3 = 0 ,

say by Gaussian elimination, to compute that n is any multiple of (3, ˜7, 1).

Finally, use n and p to write out the point-normal form

I 3(x - 2) - 7(y 1) + l(z - 1) = 0

3x ˜ 7y + z = 0.

or

Hyperplanes

A line in R2 and a plane in RвЂ™ are examples of sets described by a single linear

equation in RвЂќ. Such spaces are often called hyperplanes. A line in R2 can be

written as

a,x, + azx? = d,

and a plane in RвЂќ can be written in point-normal form as

Similarly. a hyperplane in RвЂќ can be written in point-normal form as

a,˜, + apx> + -t a,.˜,, = d. (19)

The hyperplane described by equation (IO) can be thought of as the set of all vectors

with tail at (0, , 0, d/a,r) which are perpendicular to the vector n = (a,, , a.).

We continue to call n a normal vector to the hyperplane.

EXERCISES

IO.32 Dors the point

10.33 Derive parametric and nonparametric equations for the lines which pass through

each of the following pairs of points in RI:

b) (1. I) and (4, IO):

u) (I, 2) and (3,6); c) (3, 0) and (0, 4).

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PLANES

10.34 Writs the parametric equations for each of the following lines and planes:

b) 3x, + 4x2 = 12;

a) x2 = 3x, - 7;

d) x, 2x2 + 3x1 = 6.

c) x, + XI + xi = 3;

10.35 Write nonparametric equations for each of the following lines and planes:

a) x = 3 4t, )вЂ˜=1+2t;

b) x = 21, y=,+t;

c)x=l+s+r, y = 2 + 3s + 4r, z = s - 1;

d) * = 2 - 3s + t, y = 4, z = 1 + s + l.

10.36 Derive parametric and nonparametric equations for the planes through each of the

following triplets of points in R3:

a) (6, 0, 01, (0, -6, 01, (4 4 3);

h) (0,3,2), (3,3, 11, (2 5,O).

10.37 Nonparametric equations of a line in RвЂ™ arc equations of the form

These are called symmetric equations of the line. They can be derived from the

parametric equations hy eliminating f, just as one does in the plane.

a) What are the parametric equations which correspond to the symmetric equations

(20)?

h) In form (20) one can view the line as the intersection of which two planes?

c) Find the symmetric equations of the following two lines in RвЂ™:

i ) xl =2-l ii) x, = I + 4f

XL = 3 + 4r II = 2 + 51

xi = 3 + hr.

x1 = 1 + jr:

d) Fur each line in part r. find the equations of two planes whose intersection is

that line.

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