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Chapter 10. Vectors and the Geometry of Space
- 10장 답안입니다. 자세한 답안이 아닌 학습 시 정답을 확인하기 위한 간략한 답안입니다. 시험에서는 자세한
§10.3. The Dot product
- Find ∙ . (a) (b)
(c) , the angle between and is
sol) (a) × × (b) × × × (c) × × cos
- Find the scalar and vector projection of onto .
(a) ^ , ^ ^ (b) ^
sol) (a)
× ×
(b) (^)
^
× × ×
- Show that the vector or
is orthogonal to .
sol) ∙ or
∙ ∙ ^
. So, it is orthogonal.
4. Use a scalar projection to show that the distance from a point
(^) to the line
is
sol) Let be the line , and let be the coordinates of a fixed point on . For any
point on the line , . We can write ∙ . So, any point
on the line is given by the above equation. It follows that is orthogonal to any vector
parallel to the line . If we say and
, then we can get
^
^ ^
.
Since lies on ,
^ ^
.
- If ,
and
, show that the vector equation
∙ represents a sphere and find its center and radius.
sol) ∙
Hence the given vector equation is a sphere with center ^
(^) and radius
.
- Show that if and are orthogonal, then the vectors and must have the same length.
sol) Since and are orthogonal, ∙ ^
, so that
.
∴ and must have the same length.
Then the area of the parallelogram with vertices is ^
×
^
. Thus
×
^
(b)
7. (a) Let be a point not on the plane that passes through the points , , and . Show that the
distance from the point to the plane is ^
×
×
^
where
^
^
^
(b) Use the formula in part (a) to find the distance from the point ^ to the plane through the
points , ^ and .
(sol) (a) By the formula 14 in the textbook, (p.837), the vomule of the parallelepiped determined by
the vector
, and
is
⋅ ^
×
. On the other hand, the volume is^
×
(^) (Consider the
following picture). So ^
×
×
^
(b)
- Suppose that
(a) If
, does it follow that
(b) If
×
^
×
, does it follow that
^
(c) If
^
and
×
^
×
, does it follow that
^
(sol) (a) (False) There exists a non-zero vector
st^
^ If we choose^
and^
, then
but
(b) (False) Let
and
, then
×
^
×
but
(c) (True) Suppose
× ^
^
^
and
^
. Then
^
^
and
∙ ^
^
Thus ^
^
^
^ and hence ^
- Determine whether the statement is true or false. If it is true, give a brief reason. If it is false,
give a counter example or disprove it.
(a) For any vectors
×
(b) For any vectors
^
×
(c) For any vectors
^
, ^
≤ ^
^
(sol)
(a) (True)
×
sin ,^ ^
cos
×
sin
cos
(b) (True)
×
is orthogonal to
. Thus
×
(c) (True)
^
^
cos , Since ^
≥ , ^
≥ and cos ≥ ,
⇒ ^
^
^
cos ≤ ^
^
- (a) Prove that
×
×
(sol) (a) Let
〈
〉
〈
〉
〈
〉
×
×
〈
〉
〈
〉
〈
〉
^
^
^
(b) Use (a) to prove that
×
×
×
×
×
×
(sol)
×
×
×
×
×
×
(c) Prove that
×
×
(sol) By Property 5 of Theorem 11 (p.836) and (a), we have
×
∙ ^
×
×
×
^
⋅ ^
^
^
^
^
^
^
^
- Show that the line , , and the plane are parallel,
and find the distance between them.
(Sol) Note that the point on the line does not lie in the plane. So, to show that they are
parallel, it is enough to show that the direction vector of the line is perpendicular to the normal vector
of the plane.
Since ⋅ , the vectors are perpendicular to each other.
To find the distance, choose a point that lies on the line.
The distance between and the plane is
^
and it is equal to the distance between the line and the plane.
- Find an equation of the plane through the point ^ and parallel to the vectors i k^ and j k
(Sol) Since the plane is parallel to both i k^ and j k^ , i k × j k k j i
is a normal vector of the plane. Therefore, the plane is or
§ 10.6. Cylinders and quadric surfaces
- Reduce the equation to one of the standard forms, classify the surface, and sketch it.
(a)
.
⇒
, an ellipsoid with center
(b)
.
⇒
, an elliptic paraboloid with vertex and
axis the horizontal line , .
