James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Chapter 12
Concept Check Answers (continued)
10. (a) How do you find the area of the parallelogram determined
by a and b?
The area of the parallelogram determined by a and b is
the length of the cross product: | a 3 b | .
(b) How do you find the volume of the parallelepiped
determined by a, b, and c?
The volume of the parallelepiped determined by a, b,
and c is the magnitude of their scalar triple product:
| a sb 3 cd | .
11. How do you find a vector perpendicular to a plane?
If an equation of the plane is known, it can be written in
the form ax 1 by 1 cz 1 d − 0. A normal vector, which is
perpendicular to the plane, is ka, b, cl (or any nonzero scalar
multiple of ka, b, cl). If an equation is not known, we can use
points on the plane to find two nonparallel vectors that lie in
the plane. The cross product of these vectors is a vector perpendicular
to the plane.
12. How do you find the angle between two intersecting planes?
The angle between two intersecting planes is defined as the
acute angle between their normal vectors. If n 1 and n 2 are
the normal vectors, then
cos −
n1 n2
| n 1 || n 2 |
13. Write a vector equation, parametric equations, and sym metric
equations for a line.
A vector equation for a line that is parallel to a vector v
and that passes through a point with position vector r 0 is
r − r 0 1 tv. Parametric equations for a line through the point
sx 0, y 0, z 0d and parallel to the vector ka, b, cl are
x − x 0 1 at y − y 0 1 bt z − z 0 1 ct
while symmetric equations are
x 2 x 0
− y 2 y0
a b
− z 2 z0
c
14. Write a vector equation and a scalar equation for a plane.
A vector equation of a plane that passes through a point with
position vector r 0 and that has normal vector n (meaning n is
orthogonal to the plane) is n sr 2 r 0d − 0 or, equivalently,
n r − n r 0.
A scalar equation of a plane through a point sx 0, y 0, z 0d with
normal vector n − ka, b, c l is
asx 2 x 0d 1 bsy 2 y 0d 1 csz 2 z 0d − 0
15. (a) How do you tell if two vectors are parallel?
Two (nonzero) vectors are parallel if and only if one is a
scalar multiple of the other. In addition, two nonzero vectors
are parallel if and only if their cross product is 0.
(b) How do you tell if two vectors are perpendicular?
Two vectors are perpendicular if and only if their dot
product is 0.
(c) How do you tell if two planes are parallel?
Two planes are parallel if and only if their normal vectors
are parallel.
16. (a) Describe a method for determining whether three points
P, Q, and R lie on the same line.
Determine the vectors PQ l − a and PR l − b. If there is a
scalar t such that a − tb, then the vectors are parallel and
the points must all lie on the same line.
Alternatively, if PQ l 3 PR l − 0, then PQ l and PR l are
parallel, so P, Q, and R are collinear.
An algebraic method is to determine an equation of the
line joining two of the points, and then check whether or
not the third point satisfies this equation.
(b) Describe a method for determining whether four points
P, Q, R, and S lie in the same plane.
Find the vectors PQ l − a, PR l l
− b, PS − c. Then a 3 b
is normal to the plane formed by P, Q, and R, and so S
lies on this plane if a 3 b and c are orthogonal, that is, if
sa 3 bd c − 0.
Alternatively, we can check if the volume of the parallelepiped
determined by a, b, and c is 0 (see Example
12.4.5).
An algebraic method is to find an equation of the plane
determined by three of the points, and then check whether
or not the fourth point satisfies this equation.
17. (a) How do you find the distance from a point to a line?
Let P be a point not on the line L that passes through the
points Q and R and let a − QR l , b − QP l . The distance
from the point P to the line L is
d − | a 3 b |
| a |
(b) How do you find the distance from a point to a plane?
Let P 0sx 0, y 0, z 0d be any point in the plane
ax 1 by 1 cz 1 d − 0 and let P 1sx 1, y 1, z 1d be a point not
in the plane. If b − P¸P¡ A − kx 1 2 x 0, y 1 2 y 0, z 1 2 z 0l,
then the distance D from P 1 to the plane is equal to the
absolute value of the scalar projection of b onto the
plane’s normal vector n − ka, b, cl:
D − | comp n b | − | n b |
| n |
− | ax1 1 by1 1 cz1 1 d |
sa 2 1 b 2 1 c 2
(c) How do you find the distance between two lines?
Two skew lines L 1 and L 2 can be viewed as lying on two
parallel planes, each with normal vector n − v 1 3 v 2,
where v 1 and v 2 are the direction vectors of L 1 and L 2.
After choosing one point on L 1 and determining the equation
of the plane containing L 2, we can proceed as in
part (b). (See Example 12.5.10.)
(continued)
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