Samenvatting Wiskunde II
Vectors:
𝑢1 +𝑣1 𝑐𝑢1
𝑢
⃗ + 𝑣 = ( 𝑢2+𝑣
. 2
) ∈ ℝ𝑛 ⃗ = (𝑐𝑢. 2 ) ∈ ℝ𝑛
𝑐𝑢 𝑣−𝑢
⃗ = 𝑣 + (−𝑢
⃗)
. .
𝑢𝑛 +𝑣𝑛 𝑐𝑢𝑛
Properties:
4. 𝑢
⃗ +𝑣 =𝑣+𝑢 ⃗ ⃗ + 𝑣) = 𝑐𝑢
1. 𝑐(𝑢 ⃗ + 𝑐𝑣
5. ⃗ + (𝑣 + 𝑤
𝑢 ⃗⃗ ) = (𝑢
⃗ + 𝑣) + 𝑤
⃗⃗ 2. (𝑐 + 𝑑)𝑢
⃗ = 𝑐𝑢
⃗ + 𝑑𝑢
⃗
6. 𝑢
⃗ +0⃗ =𝑢⃗ 3. (𝑐𝑑)𝑢
⃗ = 𝑐(𝑑𝑢⃗)
7. 𝑢
⃗ + (−𝑢⃗)=0 ⃗
The length of a vector 𝑢 ⃗ ∈ ℝ𝑛 is defined by ‖𝑢 ⃗ ‖ = √𝑢
⃗ ∙𝑢⃗ = √𝑎2 + 𝑏 2 . In ℝ𝑛 there are n
unit vectors 𝑒1 , 𝑒2 , … , 𝑒𝑛−1 , 𝑒𝑛 with length 1. Take an arbitrary vector 𝑣 ≠ ⃗0, then the vector
1
𝑢
⃗ = ‖𝑣⃗‖ 𝑣 has length 1.
Vector 𝑣 is a linear combination of the vectors 𝑣1 , 𝑣2 , … , 𝑣𝑘 , if there are scalars
𝑐1 , 𝑐2 , … , 𝑐𝑘 ∈ ℝ such that: 𝑣 = 𝑐1 𝑣1 , 𝑐2 𝑣2 , … , 𝑐𝑘 𝑣𝑘 .
Inner-product/dot-product: ⃗ ∙ 𝑣 = 𝑢1 𝑣1 + 𝑢2 𝑣2 + ⋯ + 𝑢𝑛 𝑣𝑛 = ∑𝑛𝑖=1 𝑢𝑖 𝑣𝑖 ∈ ℝ
𝑢
Properties inner-product:
1. 𝑢⃗ ∙𝑣 =𝑣∙𝑢 ⃗ 4. 𝑢
⃗ ∙𝑣 ≥0
2. 𝑢 (𝑣
⃗ ∙ +𝑤 )
⃗⃗ = 𝑢 ⃗ ∙𝑣+𝑢
⃗ ∙𝑤
⃗⃗ 5. 𝑢
⃗ ∙𝑢⃗ =0 ↔ 𝑢 ⃗ =0 ⃗
3. (𝑐𝑢 ⃗ ) ∙ 𝑣 = 𝑐(𝑢⃗ ∙ 𝑣)
⃗ , 𝑣 ∈ ℝ2 and let 𝜃 = ∠(𝑢
Suppose 𝑢 ⃗ , 𝑣) ∈ [0, 𝜋] be the angle between these vectors.
⃗⃗⃗
⃗ ∙𝑣
𝑢 ⃗
Then: cos(𝜃) =
‖𝑢
⃗ ‖‖𝑣
⃗‖
⃗ , 𝑣 ∈ ℝ𝑛 are perpendicular if 𝑢
The two vectors 𝑢 ⃗ ∙ 𝑣 = 0. Notation 𝑢
⃗ ⊥ 𝑣.
⃗ , 𝑣 ∈ ℝ𝑛 . Then it holds |𝑢
Cauchy-Schwarz: Let 𝑢 ⃗ ∙ 𝑣| ≤ ‖𝑢
⃗ ‖‖𝑣‖.
⃗ , 𝑣 ∈ ℝ𝑛 . Then ‖𝑢
Triangle inequality: Let 𝑢 ⃗ + 𝑣 ‖ ≤ ‖𝑢
⃗ ‖ + ‖𝑣‖
Lines and Planes
Summary: Normal form Parametric equation / Point normal equation
vector representation
Line in ℝ2 𝑎𝑥 + 𝑏𝑦 = 𝑐 𝑥 = 𝑝 + 𝑡𝑑 𝑛⃗ ∙ (𝑥 − 𝑝) = 0
Line in ℝ3 𝑎 𝑥 + 𝑏1 𝑦 + 𝑐1 𝑧 = 𝑑1 𝑥 = 𝑝 + 𝑡𝑑 𝑛 ∙ (𝑥
⃗⃗⃗⃗ ⃗⃗⃗1 − 𝑝⃗⃗⃗1 ) = 0
{ 1 { 1
𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 𝑧 = 𝑑2 𝑛2 ∙ (𝑥
⃗⃗⃗⃗ ⃗⃗⃗⃗2 − 𝑝
⃗⃗⃗⃗2 ) = 0
3
Plane in ℝ 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 𝑥 = 𝑝 + 𝑡𝑑 + 𝑠𝑢 ⃗ 𝑛⃗ ∙ (𝑥 − 𝑝) = 0
𝑎
𝑎
Normal of line in ℝ2 = ⃗⃗⃗⃗
𝑁𝑙 = ( ) = 𝑛⃗, normal of line/plane in ℝ3 = 𝑛⃗ = (𝑏)
𝑏
𝑐
Direction vector = 𝑑 , normal is orthogonal to the direction vector (cylinder - Line in ℝ3 )!
1
Vectors:
𝑢1 +𝑣1 𝑐𝑢1
𝑢
⃗ + 𝑣 = ( 𝑢2+𝑣
. 2
) ∈ ℝ𝑛 ⃗ = (𝑐𝑢. 2 ) ∈ ℝ𝑛
𝑐𝑢 𝑣−𝑢
⃗ = 𝑣 + (−𝑢
⃗)
. .
𝑢𝑛 +𝑣𝑛 𝑐𝑢𝑛
Properties:
4. 𝑢
⃗ +𝑣 =𝑣+𝑢 ⃗ ⃗ + 𝑣) = 𝑐𝑢
1. 𝑐(𝑢 ⃗ + 𝑐𝑣
5. ⃗ + (𝑣 + 𝑤
𝑢 ⃗⃗ ) = (𝑢
⃗ + 𝑣) + 𝑤
⃗⃗ 2. (𝑐 + 𝑑)𝑢
⃗ = 𝑐𝑢
⃗ + 𝑑𝑢
⃗
6. 𝑢
⃗ +0⃗ =𝑢⃗ 3. (𝑐𝑑)𝑢
⃗ = 𝑐(𝑑𝑢⃗)
7. 𝑢
⃗ + (−𝑢⃗)=0 ⃗
The length of a vector 𝑢 ⃗ ∈ ℝ𝑛 is defined by ‖𝑢 ⃗ ‖ = √𝑢
⃗ ∙𝑢⃗ = √𝑎2 + 𝑏 2 . In ℝ𝑛 there are n
unit vectors 𝑒1 , 𝑒2 , … , 𝑒𝑛−1 , 𝑒𝑛 with length 1. Take an arbitrary vector 𝑣 ≠ ⃗0, then the vector
1
𝑢
⃗ = ‖𝑣⃗‖ 𝑣 has length 1.
Vector 𝑣 is a linear combination of the vectors 𝑣1 , 𝑣2 , … , 𝑣𝑘 , if there are scalars
𝑐1 , 𝑐2 , … , 𝑐𝑘 ∈ ℝ such that: 𝑣 = 𝑐1 𝑣1 , 𝑐2 𝑣2 , … , 𝑐𝑘 𝑣𝑘 .
Inner-product/dot-product: ⃗ ∙ 𝑣 = 𝑢1 𝑣1 + 𝑢2 𝑣2 + ⋯ + 𝑢𝑛 𝑣𝑛 = ∑𝑛𝑖=1 𝑢𝑖 𝑣𝑖 ∈ ℝ
𝑢
Properties inner-product:
1. 𝑢⃗ ∙𝑣 =𝑣∙𝑢 ⃗ 4. 𝑢
⃗ ∙𝑣 ≥0
2. 𝑢 (𝑣
⃗ ∙ +𝑤 )
⃗⃗ = 𝑢 ⃗ ∙𝑣+𝑢
⃗ ∙𝑤
⃗⃗ 5. 𝑢
⃗ ∙𝑢⃗ =0 ↔ 𝑢 ⃗ =0 ⃗
3. (𝑐𝑢 ⃗ ) ∙ 𝑣 = 𝑐(𝑢⃗ ∙ 𝑣)
⃗ , 𝑣 ∈ ℝ2 and let 𝜃 = ∠(𝑢
Suppose 𝑢 ⃗ , 𝑣) ∈ [0, 𝜋] be the angle between these vectors.
⃗⃗⃗
⃗ ∙𝑣
𝑢 ⃗
Then: cos(𝜃) =
‖𝑢
⃗ ‖‖𝑣
⃗‖
⃗ , 𝑣 ∈ ℝ𝑛 are perpendicular if 𝑢
The two vectors 𝑢 ⃗ ∙ 𝑣 = 0. Notation 𝑢
⃗ ⊥ 𝑣.
⃗ , 𝑣 ∈ ℝ𝑛 . Then it holds |𝑢
Cauchy-Schwarz: Let 𝑢 ⃗ ∙ 𝑣| ≤ ‖𝑢
⃗ ‖‖𝑣‖.
⃗ , 𝑣 ∈ ℝ𝑛 . Then ‖𝑢
Triangle inequality: Let 𝑢 ⃗ + 𝑣 ‖ ≤ ‖𝑢
⃗ ‖ + ‖𝑣‖
Lines and Planes
Summary: Normal form Parametric equation / Point normal equation
vector representation
Line in ℝ2 𝑎𝑥 + 𝑏𝑦 = 𝑐 𝑥 = 𝑝 + 𝑡𝑑 𝑛⃗ ∙ (𝑥 − 𝑝) = 0
Line in ℝ3 𝑎 𝑥 + 𝑏1 𝑦 + 𝑐1 𝑧 = 𝑑1 𝑥 = 𝑝 + 𝑡𝑑 𝑛 ∙ (𝑥
⃗⃗⃗⃗ ⃗⃗⃗1 − 𝑝⃗⃗⃗1 ) = 0
{ 1 { 1
𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 𝑧 = 𝑑2 𝑛2 ∙ (𝑥
⃗⃗⃗⃗ ⃗⃗⃗⃗2 − 𝑝
⃗⃗⃗⃗2 ) = 0
3
Plane in ℝ 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 𝑥 = 𝑝 + 𝑡𝑑 + 𝑠𝑢 ⃗ 𝑛⃗ ∙ (𝑥 − 𝑝) = 0
𝑎
𝑎
Normal of line in ℝ2 = ⃗⃗⃗⃗
𝑁𝑙 = ( ) = 𝑛⃗, normal of line/plane in ℝ3 = 𝑛⃗ = (𝑏)
𝑏
𝑐
Direction vector = 𝑑 , normal is orthogonal to the direction vector (cylinder - Line in ℝ3 )!
1
