# Chapter 9 Vectors and the Geometry of Space (向量與空間幾何)

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## Vectors and the Geometry of Space(向量與空間幾何)

### Hung-Yuan Fan (范洪源)

Department of Mathematics, National Taiwan Normal University, Taiwan

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## 本章預定授課範圍

### 9.7 Cylindrical and Spherical Coordinates

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 2/60

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## Vectors in the Plane (平面向量)

Two-Dimensional Euclidean (Vector) Space R2 ={(v1, v2)| v1, v2∈ R}

={v = ⟨v1, v2⟩ | v is a vector (向量)}

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 4/60

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## Definitions and Notations (1/2)

v + u =⟨v1, v2⟩ + ⟨u1, u2⟩ = ⟨v1+ u1, v2+ u2⟩ ∀ v, u ∈ R2. Scalar Multiplication (純量乘法):

cv = c⟨v1, v2⟩ = ⟨cv1, cv2⟩ ∀ c ∈ R and v ∈ R2. Length or Norm (範數) of a Vector:

∥v∥ = ∥⟨v1, v2⟩∥ =q

v21+ v22 ≥ 0 ∀ v ∈ R2.

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## Definitions and Notations (2/2)

If i =⟨1, 0⟩ and j = ⟨0, 1⟩ are standard unit vectors in R2, then v =⟨v1, v2⟩ = v1i + v2j ∀ v ∈ R2.

If v is represented by the directed line segment from P(p1, p2) to Q(q1, q2), then it has the component form (分量形式)

v =⟨v1, v2⟩ = ⟨q1− p1, q2− p2⟩ ∈ R2.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 6/60

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## Vectors in Space (空間向量)

Three-Dimensional Euclidean (Vector) Space R3 ={(v1, v2, v3)| v1, v2, v3 ∈ R}

={v = ⟨v1, v2, v3⟩ | v is a vector in space}

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## Vectors in Space

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 8/60

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## (1/2)

v + u =⟨v1, v2, v3⟩ + ⟨u1, u2, u3

=⟨v1+ u1, v2+ u2, v3+ u3⟩ ∀ v, u ∈ R3. Scalar Multiplication (純量乘法):

cv = c⟨v1, v2, v3⟩ = ⟨cv1, cv2, cv3⟩ ∀ c ∈ R and v ∈ R3. Length or Norm of a Vector:

∥v∥ = ∥⟨v1, v2, v3⟩∥ =q

v21+ v22+ v23 ≥ 0 ∀ v ∈ R3.

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## (2/2)

If i =⟨1, 0, 0⟩, j = ⟨0, 1, 0⟩ and k = ⟨0, 0, 1⟩ are standard unit vectors in R3, then

v =⟨v1, v2, v3⟩ = v1i + v2j + v3k ∀ v ∈ R3.

If v is represented by the directed line segment from

P(p1, p2, p3) to Q(q1, q2, q3), then it has the component form v =⟨v1, v2, v3⟩ = ⟨q1− p1, q2− p2, q3− p3⟩ ∈ R3.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 10/60

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## 向量 −→PQ 的示意圖 (承上頁)

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 12/60

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## The Dot Product of Vectors (向量內積)

### Def. (向量內積的定義)

(1) The dot product of u =⟨u1, u2⟩ and v = ⟨v1, v2⟩ is u• v = u1v1+ u2v2 ∈ R.

(2) The dot product of u =⟨u1, u2, u3⟩ and v = ⟨v1, v2, v3⟩ is u• v = u1v1+ u2v2+ u3v3 ∈ R.

(3) In some textbooks, the dot product is also called the inner product of vectors.

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 14/60

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### Some Special Vectors

Let u and v be vectors inR2 or R3.

∥v∥v is the unit vector in the direction of v̸= 0.

(沿著 v 方向的單位向量)

u and v are parallel vectors (平行向量) if ∃ c ∈ R s.t. u = cv. u and v are orthogonal vectors (垂直向量) if u• v = 0.

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 16/60

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## Cross Product of Vectors in R

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### Def. (空間向量的外積)

The cross product of u =⟨u1, u2, u3⟩ and v = ⟨v1, v2, v3⟩ is

u× v =

i j k

u1 u2 u3 v1 v2 v3

(對第一列作行列式降階!)

=

u2 u3 v2 v3

i−

u1 u3 v1 v3

j +

u1 u2 v1 v2

k.

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### Notes

u× v is a vector in R3, but u• v is a scalar.

(u× v) • u = 0 = (u × v) • v, i.e., the vector u× v is orthogonal to u and v, respectively.

v× u = −(u × v), i.e., they are parallel vectors, but in the opposite directions.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 18/60

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## 外積的示意圖 (承上頁)

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 20/60

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## (空間中的曲面)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 22/60

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### Type I: Cylindrical Surfaces (柱狀曲面或是柱面)

If the line L is not parallel to the plane containing a curve C, then S = {ℓ | ℓ is a line parallel to L and intersecting C}

is a cylindrical surface, or simply a cylinder.

C is the generating curveof S.

The lines parallel to L arerulings.

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## 柱面的示意圖 (承上頁)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 24/60

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### Example

The right circular cylinder (圓柱面) with radius a > 0 is defined by S = {(x, y, z) ∈ R3| x2+ y2= a2}.

Then the generating curve of the cylinderS lying in the xy-plane is C : x2+ y2 = a2.

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## 圓柱曲面的示意圖 (承上頁)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 26/60

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## Example 1 的示意圖

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 28/60

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### Type II: Quadratic Surfaces (二次曲面)

The general equation of a quadratic surface is

Ax2+ By2+ Cz2+ Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0, where the coeﬀicients A, B,· · · , J are real numbers.

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(1) Ellipsoid: (橢圓曲面)

x2 a2 +y2

b2 +z2

c2 = 1 with a, b, c > 0.

Them the xy-trace, xz-trace and yz-trace of the surface are x2

a2 + y2

b2 = 1, x2 a2 +z2

c2 = 1 and y2 b2 +z2

c2 = 1.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 30/60

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## Ellipsoid 的示意圖 (承上頁)

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 32/60

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## Example 4 的示意圖

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(2) Hyperboloid of One Sheet: (單片雙曲面) x2

a2 +y2 b2 −z2

c2 = 1 with a, b, c > 0.

Them the xy-trace, xz-trace and yz-trace of the surface are x2

a2 + y2

b2 = 1, x2 a2 −z2

c2 = 1 and y2 b2 −z2

c2 = 1.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 34/60

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## Hyperboloid of One Sheet 的示意圖

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(3) Hyperboloid of Two Sheets: (雙片雙曲面) z2

c2 x2 a2 −y2

b2 = 1 with a, b, c > 0.

Them the xz-trace and yz-trace of the surface are z2

c2 x2

a2 = 1 and z2 c2 y2

b2 = 1, but no xy-trace!

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 36/60

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## Hyperboloid of Two Sheets 的示意圖

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 38/60

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## Example 2 的示意圖

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(4) Elliptic Cone: (橢圓錐面)

x2 a2 + y2

b2 z2

c2= 0 with a, b, c > 0.

Them the xy-trace, xz-trace and yz-trace of the surface are x2

a2 + y2

b2 = k for some k≥ 0, x2

a2 −z2

c2 = 0 =⇒z =±c axand y2

b2 −z2

c2 = 0 =⇒z =±c by.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 40/60

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## Elliptic Cone 的示意圖

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(5) Elliptic Paraboloid: (橢圓拋物面) z = x2

a2 +y2

b2 with a, b > 0.

Them the xy-trace, xz-trace and yz-trace of the surface are x2

a2 + y2

b2 = k for some k≥ 0, z = x2

a2 and z = y2 b2.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 42/60

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## Elliptic Paraboloid 的示意圖

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 44/60

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## Example 3 的示意圖

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(6) Hyperbolic Paraboloid: (雙曲拋物面) z = y2

b2 x2

a2 with a, b > 0.

Them the xy-trace, xz-trace and yz-trace of the surface are y =±b

ax, z =−x2

a2 and z = y2 b2.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 46/60

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## Cylindrical and Spherical Coordinates(柱面坐標與球面坐標)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 48/60

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### Def. (柱面坐標系統)

In the cylindrical coordinate system, a point P(x, y, z)∈ R3 is represented by an ordered triple (r, θ, z) with

(r, θ) is the polar coordinates of the (orthogonal) projection P0(x, y, 0) of P in the xy -plane.

z is the directed distance from P0 to P.

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## 柱面坐標的示意圖 (承上頁)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 50/60

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 52/60

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### Def. (球面坐標系統)

In the spherical coordinate system, a point P(x, y, z)∈ R3 is represented by an ordered triple (ρ, θ, ϕ) with

ρ =|OP| =p

x2+ y2+ z2≥ 0.

θ is the directed angle from the positive x-axis to OP0, where P0(x, y, 0) is the (orthogonal) projection of P in the xy -plane.

ϕ is the directed angle from the positive z-axis to OP.

(0≤ ϕ ≤ π)

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 54/60

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## 球面坐標的示意圖 (承上頁)

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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 56/60

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## Solutions of Example 5

(a) Since ρ2 = x2+ y2+ z2 and z = ρ cos ϕ for 0≤ ϕ ≤ π, it follows that

x2+ y2= z2=⇒ (x2+ y2+ z2)− 2z2= 0

=⇒ ρ2− 2ρ2cos2ϕ = 0 =⇒ 1 − 2 cos2ϕ = 0

=⇒ cos ϕ = ± 1

2 =⇒ ϕ = π

4 or ϕ = 4 . (b) Since ρ2 = x2+ y2+ z2 and z = ρ cos ϕ, it follows that

x2+ y2+ z2− 4z = 0 =⇒ ρ2− 4ρ cos ϕ = 0

=⇒ ρ(ρ − 4 cos ϕ) = 0

=⇒ ρ = 4 cos ϕ for 0≤ ϕ ≤ π 2.

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 58/60

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## Thank you for your attention!

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 60/60

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