Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

140 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
(c) To identify the traces in y = mx we substitute y = mx into the equation of the ellipsoid:
As expected, this is a family of ellipses.
x 2 (mx? . z 2
(6378.137) 2 + (6378.137)2 + (6356.523) 2 = 1
(1 + m 2 )x 2 z 2
(6378.137) 2 + (6356.523)2 = 1
x2
(6378.137) 2 /(1 + m2) + (6356.523)2 = 1
z2
49. If (a, b, c) satisfies z = y 2 - x 2 , then c = b 2 - a 2 . L 1 : x = a + t, y = b + t, z = c + 2( b - a )t,
£2: x =a+ t, y = b- t, z = c- 2(b + a)t. Substitute the parametric equations of £1 into the equation
of the hyperbolic paraboloid in order to find the points of intersection: z = y 2 - x 2 =>
. '
I
c + 2(b - a)t = (b + t) 2 - (a+ t) 2 = b 2 - a 2 + 2(b- a)t => c = b 2 - a 2 . As this is true for· all values oft,
L 1 lies on z = y 2 - x 2 . Performing similar operations with £2 gives: z = y 2 - x 2 =>
c- 2(b + a)t = (b- t) 2 - (a+ t? = b 2 - a 2 - 2(b + a)t => c = b 2 - a 2 . This tells us that all of £2 also lies on
51.
The curve of intersection looks like a bent ellipse. The projection
of this curve onto the xy-plane is the set of points (x, y, 0) which
satisfy x 2 + y 2 = 1 - y 2 x 2 + 2y 2 = 1
y2
x 2 + = 1. This is an equation of an ellipse.
{1//2)2
12 Review
CONCEPT CH ECK
1. A scalar is a real number, while a vector is a quantity that has both a real-valued magnitude and a direction.
2. To add two vectors geometrically, we can use either.the Triangle Law or the Parallelogram Law, as illustrated in Figures 3
and 4 in ·section 12.2. Algebraically, we add the corresponding components of the vectors.
3. For c > 0, c a is a vector with. the same direction as a and length c times the length of a. If c < 0, ca points in the opposite
direction as a and has length JcJ times the length of a. (See Figures 7 and IS in Section 12.2.) Algebraically, to find c a we
multiply each component of a by c.
4. See (I) in Section 12.2.
5. See Theorem 12.3.3 and Definition 12.3.1.
© 2012 Ccn~a.gc Lcn rnin~. All Rights Rcscr'\'ctl. Muy not be scamtcd: copied, or duplicated, or Jl(r.itcd ton publicly i1Ccessihlc website.!, in whole or In p.1rt.