Abstand zweier windschiefer Geraden

Gegeben sind zwei windschiefe Geraden: $g:{\vec {x}}={\vec {a}}+\lambda {\vec {b}}\ ;\qquad h:{\vec {x}}={\vec {c}}+\mu {\vec {d}}\ ;$

Gesucht ist der Abstand $d(g;h)=|{\overrightarrow {UV}}|$

wobei $\ U$ der Fußpunkt auf der Geraden $\ g$ und $\ V$ der Fußpunkt auf der Geraden $\ h$ bezüglich des gemeinsamen Lotes der beiden Geraden ist:

Es muss dabei für den gemeinsamen Normalenvektor ${\vec {n}}$ gelten:

${\vec {n}}\circ {\vec {b}}=0\ ;\qquad {\vec {n}}\circ {\vec {d}}=0\ ;(\ast )$

Man bildet eine geschlossene Vektorkette:

${\overrightarrow {AC}}+\mu {\vec {d}}-{\overrightarrow {UV}}-\lambda {\vec {b}}={\vec {0}}\ ;$

Mit

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 689165371
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

${\overrightarrow {UV}}=\sigma {\vec {n}}$ und nach Skalarproduktbildung mit ${\vec {n}}$ auf beiden Seiten der Gleichung ergibt sich:

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 797645118
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

${\overrightarrow {AC}}\circ {\vec {n}}+\mu {\vec {d}}\circ {\vec {n}}-\sigma {\vec {n}}\circ {\vec {n}}-\lambda {\vec {b}}\circ {\vec {n}}=0\ ;$

Mit

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 785630080
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

$(\ast )$ vereinfacht sich dies zu:

${\overrightarrow {AC}}\circ {\vec {n}}-\sigma {\vec {n}}\circ {\vec {n}}=0\ ;$

Daraus erhält man leicht $\sigma$ und damit

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 801571814
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

${\overrightarrow {UV}}=\sigma {\vec {n}}$ und somit auch den gesuchten Abstand $d(g;h)=|{\overrightarrow {UV}}|$

Beispiel:

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 790377280
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

$g:{\vec {x}}={\begin{pmatrix}0\\-2\\2\end{pmatrix}}+\lambda {\begin{pmatrix}4\\3\\-2\end{pmatrix}}\ ;\qquad h:{\vec {x}}={\begin{pmatrix}2\\4\\1\end{pmatrix}}+\mu {\begin{pmatrix}0\\1\\2\end{pmatrix}}\ ;$

${\overrightarrow {AC}}={\begin{pmatrix}2\\6\\-1\end{pmatrix}}\ ;\qquad {\vec {n}}={\begin{pmatrix}2\\-2\\1\end{pmatrix}}\ ;$

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 784127598
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

${\overrightarrow {AC}}\circ {\vec {n}}=4-12-1=-9\ ;$

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 793191224
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

${\vec {n}}\circ {\vec {n}}=4+4+1=9\ ;$

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 772930232
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

${\overrightarrow {AC}}\circ {\vec {n}}-\sigma {\vec {n}}\circ {\vec {n}}=-9-9\sigma \ ;$

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 775613133
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

$0=-9-9\sigma \ ;$

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 793323218
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

$\sigma =-1\ ;\qquad {\overrightarrow {UV}}={\begin{pmatrix}-2\\2\\-1\end{pmatrix}}\ ;$

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp3066 cp3066, Varnish XID 772892311
Upstream caches: cp3066 int
Error: 429, Too many requests (f061ab2) at Mon, 12 Jan 2026 23:20:46 GMT


Sensitive client information

IP address: 188.64.62.105

$d(g;h)=|{\overrightarrow {UV}}|={\sqrt {9}}=3\ ;$

---

Zurück zur Mathematik-Hauptseite