Shortest distance between two skew linesदो विषमतलीय रेखाओं के बीच न्यूनतम दूरीDo skew lines ke beech shortest distance
NCERT Fig 11.6 · Class 12 Maths · Chapter 11 · interactive diagram
Parts and what they mean
- Line l_1: \vec{r} = \vec{a_1} + \lambda\vec{b_1} (रेखा l_1: \vec{r} = \vec{a_1} + \lambda\vec{b_1}) — The first skew line (drawn here on the floor).
- Line l_2: \vec{r} = \vec{a_2} + \mu\vec{b_2} (रेखा l_2: \vec{r} = \vec{a_2} + \mu\vec{b_2}) — The second skew line, higher up and crossing over l_1 without meeting it.
- S — point on l_1 with position vector \vec{a_1} (S — l_1 पर स्थिति सदिश \vec{a_1} वाला बिंदु) — The known point of l_1 from its equation.
- T — point on l_2 with position vector \vec{a_2} (T — l_2 पर स्थिति सदिश \vec{a_2} वाला बिंदु) — The known point of l_2 from its equation.
- \overrightarrow{ST} = \vec{a_2} - \vec{a_1} (\overrightarrow{ST} = \vec{a_2} - \vec{a_1}) — Joins the two known points; usually longer than the shortest distance.
- P — foot of the common perpendicular on l_1 (P — l_1 पर उभयनिष्ठ लंब का पाद) — One end of the shortest-distance segment.
- Q — foot of the common perpendicular on l_2 (Q — l_2 पर उभयनिष्ठ लंब का पाद) — The other end of the shortest-distance segment.
- Shortest distance PQ (न्यूनतम दूरी PQ) — Perpendicular to both lines, so it is along \vec{b_1}\times\vec{b_2}; d = PQ.
- Unit vector \hat{n} = \dfrac{\vec{b_1}\times\vec{b_2}}{|\vec{b_1}\times\vec{b_2}|} (मात्रक सदिश \hat{n} = \dfrac{\vec{b_1}\times\vec{b_2}}{|\vec{b_1}\times\vec{b_2}|}) — The direction of PQ (drawn at S, parallel to PQ).
- θ — angle between \overrightarrow{ST} and \overrightarrow{PQ} (θ — \overrightarrow{ST} और \overrightarrow{PQ} के बीच कोण) — PQ = ST\,|\cos\theta|: the shortest distance is the projection of ST on \hat{n}.
Original diagram drawn by Simple Learning Hub, based on NCERT Fig 11.6. Report a mistake