Three Dimensional Geometry
Class 12 Mathematics · Mathematics Part II · Chapter 5
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Accountancy
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اردو (اختیاری)
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Physics
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हिंदी (ऐच्छिक)
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Physics
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اردو (کور)
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Chemistry
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Urdu Elective
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Computer Science
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Chemistry
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Fine Art · Painting
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Biology
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Informatics Practices
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Accountancy
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Biotechnology
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Accountancy
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सृजनात्मक लेखन और अनुवाद
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Business Studies
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हिंदुस्तानी संगीत (तबला / पखावज)
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Business Studies
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Economics
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Economics
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History
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History
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History
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Geography
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Geography
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Geography
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English Elective · Kaleidoscope
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Political Science
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Political Science
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Sociology
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Sociology
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Psychology
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English Core · Flamingo
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English Core · Vistas
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Home Science
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Home Science
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Summary
This chapter extends coordinate geometry to three dimensions, describing lines and planes vectorially and in Cartesian form. A line is given by \(\vec{r}=\vec{a}+\lambda\vec{b}\), where \(\vec{a}\) is a point on it and \(\vec{b}\) its direction; in Cartesian form this is \(\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}\). The angle between two lines comes from the dot product of their direction vectors. A plane is written as \(\vec{r}\cdot\hat{n}=d\), where \(\hat{n}\) is the unit normal, or in Cartesian form \(ax+by+cz=d\). The chapter computes the angle between two planes, between a line and a plane, and the shortest distance between two skew lines. These results give a complete algebraic description of position and orientation in space.
Direction cosines and ratiosEquations of a line in spaceAngle between linesEquations of a planeShortest distance between skew lines
Key terms
Direction ratios
Numbers proportional to the direction cosines of a line.
Vector equation of a line
\(\vec{r}=\vec{a}+\lambda\vec{b}\).
Cartesian equation of a line
\(\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}\).
Equation of a plane
\(\vec{r}\cdot\hat{n}=d\) or \(ax+by+cz=d\).
Skew lines
Lines that are neither parallel nor intersecting.
Shortest distance
The perpendicular distance between two skew lines.
Extra questions & answers
Try one from this chapter
The vector equation of a line is: