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When f = 0, we are at point x; and when f = 1, we are at point y. When f lies

between 0 and 1, we are at points between x and y. Consequently, we parameterize

the line segment joining x toy as

e(x,y)={(l+t)x+ty:O5r51}.

[lo.51 LINES 2 2 5

Given two points x = (a, b) and y = (c, d) on a line J? in the plane, one can

write the parameterized equation of P as (13) вЂњI the nonparameterized equation

of1 as

X2-b= d-b (Xl - 4

c-a

One can use these two expressions to pass from the parameterized equation of a

line in the plane to the nonparameterized equation, and vice versa, by first finding

two points on the line from the given equation and using these points to find the

new equation. One can also pass directly from form (IO) to form (9) by solving

the equations in (10) fort and then setting the new equations equal to each other.

For example, in equations (11, 12)

t=xl-4 r=X2-2

and

1 1

x, - 4 p%-2

x2 = x, - 2 .

SвЂќ,

--1вЂ™ Or

1

To go the other way, just note that equation (9) is the equation of the line through

the point (0, b) in the direction (1, m).

EXERCISES

10.27 Show Ihat the midpuint of J(r y) occurs where f = 4, In other words, if z =

;x L iy, show that /Ix zI/ = Jly zIJ.

10.28 For each of the fullowing pairs of points P,, P2, write the Parametric equation of

the line through P, and pl> find the midpoint ofe(p,. p2), and sketch the line.

10.29 tsIhepвЂќint( $вЂњnthcline(i) +rt)вЂ˜?

10.30 Transform each of the f<,llawing parameterircd equations into the form (9):

x, = 4 - 2 x,=3+ f x, =3+1

a) b) 4 x2 = 5 .

XI = 3 + fir; x2 = 5 *;

10.31 Transform each of the following nonparameterired equations into the form (10):

a) 2x1 = Ix, + 5 ; b) x2 = -x, + 7; c) *, = 6

ELKXDEAN SPACES I1 01

226

10.6 PLANES

Parametric Equations

A line is one-dimensional. Intuitively, the dimension of the line is refected in the

fact that it can be described using only one parameter. Planes are two-dimensional,

and so it stands to reason that they are described by expressions with two param-

eters.

To be more concrete, let T be a plane in R3 through the origin. Let Y and w be

two vectors in T, as shown in Figure 10.27. Choose v and w so that they point in

different directions, in other words, so that neither is a scalar multiple of the other.

In this case, we say that v and w are linearly independent, a topic to be discussed

in more detail in the next chapter. For any scalars s and f, the vector sv + fw is

called a linear combination of Y and w. By our geometric interpretation of scalar

multiplication and vector addition, it should be clear that all linear combinations

of v and w lie on the plane T. In fact, if we take every linear combination of v and

w, we recover the entire plane T. The equation

provides a parameteriration of the plane T,

Figure

ltl.27

If the plane does not pass through the origin but through the point p # 0

and if v and w arc linearly indcpcndent direction vectors from p which still lie

in the plane, then as indicated in Figure 10.28. we can use the above method to

parameterize the plane as

.s: f in RвЂ™.

x = p + вЂ˜iv + fW. (14)

227

110.61 PLANES

Figure

A plane not through the origin. 10.28

Just as two points determine a line, three (non-collinear) points determine a

plane. To find the parametric equation of the plane containing the points p, q, and

r, note that we can picture q - p and r - pas displacement vectors from p which

lie on the plane. So, one parameterization of the plane is

x(.7, вЂ˜) = p + s(q - p) + t(r - p)

(15)

= (1 s ˜ t)p + sq + tr.

Compare (15) with the corresponding parameterired equation (13) of a line. From

equation (15), we see that a plane is the set of those linear combinations of three

fixed wxtors whose coefficients sum to 1:

x = f, p + r*q + t,r, 1, + f2 + f? = 1. (16)

If we further restrict the scalars f; in (16) so that they are nonnegative, we obtain

the (filled-in) triangle in R3 whose vertices are p, q, and r-the darkened region

in Figure 10.29. The numbers (t,, tz, f3) are called the barycentric coordinates

of a point in this triangle. For example, the barycentric coordinates of the vertex

pare (1, 0, 0) since 11 = I, 12 = 0, and t? = 0 in expression (16) yield the point

p. Similarly, the barycentric coordinates of the vertices q and r are (0, 1, 0) and

(O,O, I), respectively. The center of mass or centroid of this triangle is the point

whose barycentric coordinates are (l/3, l/3, I /3).

Equations such as (14) and (15) give a parameterization of a two-dimensional

plane in any Euclidean space, not just RвЂќ. For example, the two-dimensional plane

Figure

Triangle with vertices p, q, and r

10.29

through the points (I, 2, 3,4), (5, 6, 7, S), and (9, 0, 1,2) in R4 has the parametric

equations

x1 = Ii- + 5s + 9t

x* = 2r + 6s + Of

X) = 3r + IS + If

xq = 4r + 8s + zt, where I + s + f = I

Nonparametric Equations

We turn now to the nonparametric equations of aplane in R-вЂ˜. Just as with a line

in R*, a plane in R3 is completely described by giving its inclination and a point

on it. We usually express its inclination by specifying a vector n, called a normal

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