08 Vektor­geometrie

408_skript_vektorgeometrie_grundlagen.pdf 409_skript_vektorgeometrie_geraden_und_ebenen.pdf 410_skript_vektorgeometrie_skp_normalformen.pdf 411_Skript_Vektorgeometrie _vektorprodukt.pdf

1. Grundlagen der Vektorgeometrie (Vector Geometry Basics)

Vector geometry provides tools to describe and analyze geometric objects like points, lines, and planes using vectors.

1.1. Skalare vs. Vektorielle Größen (Scalars vs. Vectors)

• Skalar (Scalar): A quantity defined by a magnitude (number) and possibly a unit (e.g., temperature, distance, speed/tempo) 111111111.

• Vektoriell (Vectorial): A quantity defined by magnitude and direction (e.g., velocity, force, displacement) 2222. Represented visually by arrows3.

1.2. Vektoren: Definition und Darstellung (Vectors: Definition and Representation)

• Definition: A vector is the set of all arrows (directed line segments) having the same length and direction4. Any single arrow is a representative (Repräsentant)5. Vectors can be shifted parallelly without changing6.

• Räume (Spaces): We focus on the plane ${\mathbb{R}}^2$ and space ${\mathbb{R}}^3$7. Points are represented by coordinate tuples $(x_1,x_2)$ or $(x_1,x_2,x_3)$8.

• Notation:

  • Name: $\vec{v},\vec{w}$9.

  • From points: $\overrightarrow{PQ}$ (from point P to point Q)10.

  • Coordinates (column vector): $\vec{v}=\left(\begin{array}{c}{v}_1\\ v_2\\ (v_3)\end{array}\right)$ representing the displacement in each coordinate direction11111111. The parentheses indicate the third component is optional (for ${\mathbb{R}}^2$ vs ${\mathbb{R}}^3$)12.

1.3. Spezielle Vektoren und Operationen (Special Vectors and Operations)

• Nullvektor (Zero Vector): $\vec{0}=\left(\begin{array}{c}0\\ 0\\ (0)\end{array}\right)$. Represents no displacement 13. $\overrightarrow{PQ}+\overrightarrow{QP}=\vec{0}$14.

• Gegenvektor (Opposite Vector): $-\vec{v}=(-1)\vec{v}=\left(\begin{array}{c}-v_1\\ -v_2\\ (-v_3)\end{array}\right)$. Same length, opposite direction 15.

• Skalarmultiplikation (Scalar Multiplication): $t\cdot \vec{v}=\left(\begin{array}{c}t{v}_1\\ t{v}_2\\ (t{v}_3)\end{array}\right)$. Stretches ($|t|>1$) or shrinks ($|t|<1$) the vector; reverses direction if $t<0$ 16161616.

• Vektoraddition (Vector Addition): Geometrically corresponds to placing arrows head-to-tail (parallelogram rule)17171717. Coordinates are added component-wise 18:

  $\vec{v}+\vec{w}=\left(\begin{array}{c}{v}_1+w_1\\ v_2+w_2\\ (v_3+w_3)\end{array}\right)$

• Vektorsubtraktion (Vector Subtraction): Add the opposite vector: $\vec{v}-\vec{w}=\vec{v}+(-\vec{w})$ 19:

  $\vec{v}-\vec{w}=\left(\begin{array}{c}{v}_1-w_1\\ v_2-w_2\\ (v_3-w_3)\end{array}\right)$

1.4. Norm (Länge) eines Vektors (Norm/Length of a Vector)

• Definition: The norm or length $||\vec{v}||$ is calculated using the Pythagorean theorem 20202020:

  $||\vec{v}||=\sqrt{v_1^2+v_2^2(+v_3^2)}$

• Properties:

  • $||\vec{0}||[cit{e}_{s}tart]=0$21.

  • $||-\vec{v}||[cit{e}_{s}tart]=||\vec{v}||$ 22.

  • $||t\cdot \vec{v}||=|t|[cit{e}_{s}tart]\cdot ||\vec{v}||$ 23.

1.5. Ortsvektoren (Position Vectors)

• Definition: The position vector (Ortsvektor) $\overrightarrow{OP}$ of a point $P(x_1,x_2,(x_3))$ is the vector from the origin $O(0,0,(0))$ to $P$ 24. Its coordinates are the same as the point’s coordinates 25:

  $\overrightarrow{OP}=\left(\begin{array}{c}{x}_1\\ x_2\\ (x_3)\end{array}\right)$

• Connecting Vector: The vector from point $P$ to point $Q$ is found by subtracting position vectors 26262626:

  $\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=\left(\begin{array}{c}{x}_Q-x_P\\ y_Q-y_P\\ (z_Q-z_P)\end{array}\right)$

• Mittelpunkt (Midpoint): The position vector $\overrightarrow{OM}$ of the midpoint $M$ of segment $PQ$ is the average of the endpoint position vectors 27272727:

  $\overrightarrow{OM}=\overrightarrow{OP}+\frac{1}{2}\overrightarrow{PQ}=\frac{1}{2}(\overrightarrow{OP}+\overrightarrow{OQ})$

• Schwerpunkt (Centroid): The position vector $\overrightarrow{OS}$ of the centroid $S$ of a triangle $PQR$ is the average of the vertex position vectors 28:

  $\overrightarrow{OS}=\frac{1}{3}(\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR})$

---

2. Geraden und Ebenen (Lines and Planes)

2.1. Geraden (Lines)

Parametergleichung (Vector/Parametric Equation)

• Describes a line $g$ passing through point $P$ in the direction of vector ${\vec{r}}_g=\overrightarrow{PQ}$ 29.

• Form: For any point $X$ on the line, its position vector $\vec{x}$ is given by 30:

  $g:\vec{x}=\overrightarrow{OP}+t\cdot {\vec{r}}_g(t\in \mathbb{R})$

  • $\overrightarrow{OP}$ is the stützvektor (position vector of a point on the line).

  • ${\vec{r}}_g$ is the richtungsvektor (direction vector).

  • $t$ is the parameter. Varying $t$ traces out all points on the line 31313131.

• This form works in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$32323232.

Koordinatengleichung (Coordinate Equation) - Nur in ${\mathbb{R}}^2$

• Forms:

  • Linear function: $y=px+q$ (for non-vertical lines) 33.

  • General form: $ax+by+c=0$ (works for all lines, including vertical) 34.

• There’s no direct equivalent simple coordinate equation for a line in ${\mathbb{R}}^3$35.

Umwandlungen (Conversions) - in ${\mathbb{R}}^2$

• Coordinate to Vector: Find two points $P,Q$ satisfying $ax+by+c=0$. Use them to form $\overrightarrow{OP}$ and ${\vec{r}}_g=\overrightarrow{PQ}$ 36.

• Vector to Coordinate:

  1. Find the slope $p$ from the direction vector ${\vec{r}}_g=\left(\genfrac{}{}{0ex}{}{r_1}{r_2}\right)$ as $p=r_2/r_1$ (if $r_1\ne 0$) 37.

  2. Use the point $P$ from $\overrightarrow{OP}$ in $y=px+q$ to find the y-intercept $q$ 38.

  3. Rewrite in the form $ax+by+c=0$ 39.

Lagebeziehungen und spezielle Punkte (Relative Positions and Special Points)

• Punkt auf Gerade (Point on Line): A point $R$ lies on $g:\vec{x}=\overrightarrow{OP}+t{\vec{r}}_g$ if there exists a single $t\in \mathbb{R}$ such that $\overrightarrow{OR}=\overrightarrow{OP}+t{\vec{r}}_g$ 40. This leads to a system of equations (one for each coordinate) which must have a consistent solution for $t$ 41.

• Parallele Geraden (Parallel Lines): Two lines $g:\vec{x}=\overrightarrow{OP}+s{\vec{r}}_g$ and $h:\vec{x}=\overrightarrow{OQ}+t{\vec{r}}_h$ are parallel if their direction vectors are scalar multiples of each other: ${\vec{r}}_h=\lambda {\vec{r}}_g$ for some $\lambda \ne 0$ 42.

• Schnittpunkt (Intersection Point): Find $s,t$ such that $\overrightarrow{OP}+s{\vec{r}}_g=\overrightarrow{OQ}+t{\vec{r}}_h$ 43. This is a system of linear equations for $s$ and $t$44.

  • In ${\mathbb{R}}^2$: Unique solution if not parallel. No solution if parallel and distinct. Infinite solutions if identical.

  • In ${\mathbb{R}}^3$: Unique solution if they intersect. No solution if parallel and distinct OR windschief (skew) (non-parallel, non-intersecting) 45454545. Infinite solutions if identical.

• Spurpunkte (Trace Points) in ${\mathbb{R}}^3$: Intersection points with the coordinate planes ($x_1{x}_2$-plane where $x_3=0$, $x_1{x}_3$-plane where $x_2=0$, $x_2{x}_3$-plane where $x_1=0$) 46. Find the parameter $t$ by setting the corresponding coordinate in the vector equation to zero 47.

2.2. Ebenen (Planes) - Nur in ${\mathbb{R}}^3$

Parametergleichung (Vector/Parametric Equation)

• Describes a plane $E$ passing through point $P$ and spanned by two non-collinear direction vectors $\vec{u}=\overrightarrow{PQ}$ and $\vec{v}=\overrightarrow{PR}$48484848.

• Any vector $\overrightarrow{PX}$ in the plane can be written as a unique linear combination $\overrightarrow{PX}=s\vec{u}+t\vec{v}$ 49.

• Form: For any point $X$ on the plane, its position vector $\vec{x}$ is given by 50:

  $E:\vec{x}=\overrightarrow{OP}+s\cdot \vec{u}+t\cdot \vec{v}(s,t\in \mathbb{R})$

  • $\overrightarrow{OP}$ is the stützvektor.

  • $\vec{u},\vec{v}$ are the richtungsvektoren (direction/spanning vectors).

  • $s,t$ are the parameters.

Koordinatengleichung (Coordinate Equation)

• Form: A single linear equation represents a plane in ${\mathbb{R}}^3$ 51515151:

  $E:a{x}_1+b{x}_2+c{x}_3+d=0$

• Special Cases:

  • If $d=0$, the plane contains the origin52.

  • If e.g. $a=0$, the plane is parallel to the $x_1$-axis (or contains it if $d=0$) 53.

  • If e.g. $a=b=0$ (and $c\ne 0$), the plane is $c{x}_3+d=0$ or $x_3=-d/c$, parallel to the $x_1{x}_2$-plane.

Umwandlungen (Conversions)

• Coordinate to Vector: Find three non-collinear points $P,Q,R$ satisfying the coordinate equation. Use them to form $\overrightarrow{OP}$, $\vec{u}=\overrightarrow{PQ}$, $\vec{v}=\overrightarrow{PR}$54.

• Vector to Coordinate: Eliminate the parameters $s,t$ from the system of three component equations 55555555.

  • $x_1=p_1+s{u}_1+t{v}_1$

  • $x_2=p_2+s{u}_2+t{v}_2$

  • $x_3=p_3+s{u}_3+t{v}_3$

Lagebeziehungen und spezielle Punkte (Relative Positions and Special Points)

• Punkt auf Ebene (Point on Plane):

  • Coordinate Eq.: Check if the point’s coordinates satisfy the equation56.

  • Vector Eq.: Check if there exist $s,t$ such that $\overrightarrow{OU}=\overrightarrow{OP}+s\vec{u}+t\vec{v}$ 57.

• Spurpunkte (Trace Points): Intersection points with the coordinate axes58. Find by setting two coordinates to zero in the coordinate equation 59. E.g., for $S_1$ on the $x_1$-axis, set $x_2=x_3=0$ and solve for $x_1$.

  • Achsenabschnittsform (Intercept Form): If intercepts are $A,B,C$, the equation is $\frac{x_1}{A}+\frac{x_2}{B}+\frac{x_3}{C}=1$ 60.

• Schnitt Gerade-Ebene (Line-Plane Intersection): Substitute the line’s parametric component equations ($x_1=p_1+u{r}_1$, etc.) into the plane’s coordinate equation and solve for the line parameter $u$ 61. Substitute $u$back into the line equation to get the point62. (Alternatively, equate vector forms and solve the $3\times 3$ system for $s,t,u$ 63).

  • Possible outcomes: Unique intersection point, line lies within the plane (infinite solutions), line is parallel to the plane (no solution).

• Schnitt Ebene-Ebene (Plane-Plane Intersection): Solve the system of two coordinate equations 64.

  • If planes are not parallel, the solution set is a line (the Schnittgerade). Find two points that satisfy both equations (e.g., by setting $x_1=0$ and solving for $x_2,x_3$, then setting $x_1=1$ and solving again) to define the line 65656565.

  • If planes are parallel and distinct, no solution.

  • If planes are identical, infinite solutions (the plane itself).

---

3. Das Skalarprodukt (Scalar Product / Dot Product)

A way to “multiply” two vectors resulting in a scalar.

3.1. Definitionen

1. Geometrisch: Based on the angle $\alpha$ between the vectors ($0\le \alpha \le 180^{\circ }$) 666666666666666666:

   $\vec{v}\cdot \vec{w}=||\vec{v}||\cdot ||\vec{w}||\cdot \cos (\alpha )$

   • If either vector is $\vec{0}$, the dot product is 067676767.

2. Algebraisch: Based on components 68686868:

   $\vec{v}\cdot \vec{w}=v_1{w}_1+v_2{w}_2(+v_3{w}_3)$

3.2. Eigenschaften und Gesetze (Properties and Laws)

• The result is a scalar (a number), not a vector69.

• Kommutativgesetz (Commutative): $\vec{v}\cdot \vec{w}=\vec{w}\cdot \vec{v}$ 70.

• Distributivgesetz (Distributive): $\vec{u}\cdot (\vec{v}+\vec{w})=\vec{u}\cdot \vec{v}+\vec{u}\cdot \vec{w}$ 71.

• Scalar Multiplication: $(\lambda \vec{v})\cdot (\mu \vec{w})=\lambda \mu (\vec{v}\cdot \vec{w})$ 72.

• Relation to Norm: $\vec{v}\cdot \vec{v}=||\vec{v}|{|}^2$73.

3.3. Geometrische Bedeutung und Anwendungen

• Vorzeichen (Sign):

  • $\vec{v}\cdot \vec{w}>0⟺\alpha$ is acute ($<90^{\circ }$)74.

  • $\vec{v}\cdot \vec{w}<0⟺\alpha$ is obtuse ($>90^{\circ }$)75.

  • $\vec{v}\cdot \vec{w}=0⟺\alpha =90^{\circ }$ (orthogonal) or one vector is $\vec{0}$ 76.

• Orthogonalitätstest (Orthogonality Test): Non-zero vectors $\vec{v}$ and $\vec{w}$ are perpendicular if and only if $\vec{v}\cdot \vec{w}=0$ 77777777.

• Winkelberechnung (Angle Calculation): The angle $\alpha$ between non-zero vectors $\vec{v}$ and $\vec{w}$ is given by 787878787878787878:

  $\cos (\alpha )=\frac{\vec{v}\cdot \vec{w}}{||\vec{v}||\cdot ||\vec{w}||}$

• Projektion (Projection): The vector projection ${\vec{v}}^{\prime }$ of $\vec{w}$ onto the direction of $\vec{v}$ is 79:

  ${\vec{v}}^{\prime }=\left(\frac{\vec{w}\cdot \vec{v}}{||\vec{v}|{|}^2}\right)\vec{v}$

  The scalar projection (length of ${\vec{v}}^{\prime }$ with sign) is $||\vec{w}||\cos (\alpha )=\frac{\vec{w}\cdot \vec{v}}{||\vec{v}||}$.

• Arbeit (Work): Physical work $A$ done by a force $\vec{w}$ over a displacement $\vec{v}$ is $A=\vec{w}\cdot \vec{v}$ (if force is constant)808080808080808080.

---

4. Normalenvektoren und Hesse-Normalform (Normal Vectors and Hesse Normal Form)

4.1. Normalenvektor (Normal Vector)

• Definition: A vector $\vec{n}$ that is perpendicular (orthogonal) to a given line (in ${\mathbb{R}}^2$) or plane (in ${\mathbb{R}}^3$) 81. It is not unique; any scalar multiple $\lambda \vec{n}$ ($\lambda \ne 0$) is also a normal vector82.

• From Coordinate Equation:

  • For a line $g:ax+by+c=0$ in ${\mathbb{R}}^2$, ${\vec{n}}_g=\left(\genfrac{}{}{0ex}{}{a}{b}\right)$ is a normal vector 83838383.

  • For a plane $E:ax+by+cz+d=0$ in ${\mathbb{R}}^3$, ${\vec{n}}_E=\left(\begin{array}{c}a\\ b\\ c\end{array}\right)$ is a normal vector 84848484.

• Verification: Show that $\vec{n}\cdot \overrightarrow{PQ}=0$ for any two points $P,Q$ on the line/plane 85858585.

4.2. Anwendungen von Normalenvektoren

• Finding orthogonal line/plane: The normal vector of the given object becomes the direction vector of the orthogonal object 86.

• Reflections: To reflect a point $O$ across a plane $E$, find the line through $O$ normal to $E$. Find the intersection $S$ of the line and plane. The reflection $P^{\prime }$ satisfies $\overrightarrow{O{P}^{\prime }}=2\overrightarrow{OS}$ 87.

• Angle between Planes: The angle between two planes $E$ and $F$ is equal to the angle (or its supplement $180^{\circ }-\alpha$) between their normal vectors ${\vec{n}}_E$ and ${\vec{n}}_F$ 88. Calculate using the dot product formula for $\cos (\alpha )$.

4.3. Abstand Punkt-Ebene / Punkt-Gerade (Distance Point-Plane / Point-Line)

• The distance $D$ from a point $P(x_P,y_P,z_P)$ to a plane $E:ax+by+cz+d=0$ is the length of the projection of $\overrightarrow{BP}$ onto the normal vector ${\vec{n}}_E$, where $B$ is any point on the plane 89898989.

• Formula (${\mathbb{R}}^3$): 90

  $D=\left|\frac{a{x}_P+b{y}_P+c{z}_P+d}{||{\vec{n}}_E||}\right|=\left|\frac{a{x}_P+b{y}_P+c{z}_P+d}{\sqrt{a^2+b^2+c^2}}\right|$

• Formula (${\mathbb{R}}^2$): For distance from $P(x_P,y_P)$ to line $g:ax+by+c=0$ 91:

  $D=\left|\frac{a{x}_P+b{y}_P+c}{||{\vec{n}}_g||}\right|=\left|\frac{a{x}_P+b{y}_P+c}{\sqrt{a^2+b^2}}\right|$

• Interpretation of Sign: The sign of the expression inside the absolute value indicates on which side of the plane/line the point $P$ lies, relative to the direction of the normal vector $\vec{n}$ 92929292.

4.4. Hesse-Normalform (Hesse Normal Form - HNF)

• A normalized version of the coordinate equation, obtained by dividing by the norm of the normal vector, $||\vec{n}||$93.

• Form (Plane): 94

  $\frac{ax+by+cz+d}{\sqrt{a^2+b^2+c^2}}=0$

• Form (Line): 95

  $\frac{ax+by+c}{\sqrt{a^2+b^2}}=0$

• Advantage: Plugging a point $P$‘s coordinates into the left side of the HNF directly gives the signed distance from the point to the plane/line96. The absolute value is the geometric distance $D$. The constant term in the HNF gives the signed distance from the origin97.

---

5. Das Vektorprodukt (Vector Product / Cross Product)

A way to multiply two vectors in ${\mathbb{R}}^3$ resulting in another vector in ${\mathbb{R}}^3$. Does not exist in ${\mathbb{R}}^2$.

5.1. Motivation

• Physics: Lorentz force ${\vec{F}}_L=q(\vec{v}\times \vec{B})$ acts perpendicular to both velocity $\vec{v}$ and magnetic field $\vec{B}$ 98989898.

• Geometry: Need an operation to find a vector orthogonal to two given vectors (e.g., normal to a plane) 99.

5.2. Geometrische Definition (Geometric Definition)

The vector product $\vec{c}=\vec{v}\times \vec{w}$ is uniquely defined by three properties 100:

1. Orthogonalität (Orthogonality): $\vec{c}$ is perpendicular to both $\vec{v}$ and $\vec{w}$ (and thus perpendicular to the plane they span).

2. Betrag/Norm (Magnitude/Norm): $||\vec{c}||=||\vec{v}||\cdot ||\vec{w}||\cdot \sin (\alpha )$, where $\alpha$ is the angle between $\vec{v}$ and $\vec{w}$. This magnitude equals the area of the parallelogram spanned by $\vec{v}$ and $\vec{w}$ 101101101101.

3. Orientierung (Orientation): The triplet $(\vec{v},\vec{w},\vec{c})$ forms a right-handed system(Rechtssystem), following the right-hand rule 102.

• Special Case: If $\vec{v}$ and $\vec{w}$ are collinear (parallel or anti-parallel, $\alpha =0^{\circ }$ or $180^{\circ }$), then $\vec{v}\times \vec{w}=\vec{0}$103.

5.3. Algebraische Berechnung (Algebraic Calculation)

• The coordinates of $\vec{x}=\vec{v}\times \vec{w}$ are given by104104104104104104104104104:

  $\vec{v}\times \vec{w}=\left(\begin{array}{c}{v}_2{w}_3-v_3{w}_2\\ v_3{w}_1-v_1{w}_3\\ v_1{w}_2-v_2{w}_1\end{array}\right)$

• This formula ensures all three geometric properties are met 105105105105.

• Mnemonic using Determinants: The components can be remembered as $2\times 2$ determinants 106:

  $x_1=\det \left(\begin{array}{cc}{v}_2& w_2\\ v_3& w_3\end{array}\right),x_2=\det \left(\begin{array}{cc}{v}_3& w_3\\ v_1& w_1\end{array}\right),x_3=\det \left(\begin{array}{cc}{v}_1& w_1\\ v_2& w_2\end{array}\right)$

  (Note the cyclic permutation of indices 1, 2, 3 in the rows).

5.4. Eigenschaften (Properties)

• Nicht-kommutativ (Anti-commutative): $\vec{w}\times \vec{v}=-(\vec{v}\times \vec{w})$ 107.

• Nicht-assoziativ (Not associative): $(\vec{u}\times \vec{v})\times \vec{w}\ne \vec{u}\times (\vec{v}\times \vec{w})$ generally 108.

• Distributive: $\vec{u}\times (\vec{v}+\vec{w})=(\vec{u}\times \vec{v})+(\vec{u}\times \vec{w})$.

• Scalar Multiplication: $(\lambda \vec{v})\times \vec{w}=\vec{v}\times (\lambda \vec{w})=\lambda (\vec{v}\times \vec{w})$.

5.5. Geometrische Anwendungen (Geometric Applications)

• Normalenvektor einer Ebene (Plane Normal Vector): Given three non-collinear points $A,B,C$, the vector ${\vec{n}}_E=\overrightarrow{AB}\times \overrightarrow{AC}$ is normal to the plane containing them 109. This allows easy derivation of the plane’s coordinate equation $n_{1}x+n_{2}y+n_{3}z+d=0$ 110.

• Flächeninhalt (Area):

  • Area of parallelogram spanned by $\vec{v},\vec{w}$: $A_{para}=||\vec{v}\times \vec{w}||$111111111111.

  • Area of triangle $ABC$: $A_{△}=\frac{1}{2}||\overrightarrow{AB}\times \overrightarrow{AC}||$ 112112112112.

• Abstand Punkt-Gerade in ${\mathbb{R}}^3$ (Distance Point-Line in ${\mathbb{R}}^3$): The distance $x$ from a point $P$ to a line $g$ (given by point $Q$ and direction ${\vec{r}}_g$) is the height of the parallelogram spanned by $\overrightarrow{QP}$ and ${\vec{r}}_g$ 113113113113.

  $x=\frac{\text{Area of Parallelogram}}{\text{Base Length}}=\frac{||\overrightarrow{QP}\times {\vec{r}}_g||}{||{\vec{r}}_g||}$

  114