What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

Write the vector equation of a line.

\(\vec{r}=\vec{a}+\lambda\vec{b}\), where \(\vec{a}\) is the position vector of a point on the line and \(\vec{b}\) is its direction vector.

Write the Cartesian equation of a line through \((x_1,y_1,z_1)\).

\(\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}\), where \(a,b,c\) are direction ratios.

How is the angle between two lines found?

From \(\cos\theta=\dfrac{\vec{b_1}\cdot\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\), using their direction vectors.

Write the equation of a plane in normal form.

\(\vec{r}\cdot\hat{n}=d\), where \(\hat{n}\) is the unit normal and \(d\) is the distance from the origin.

When are two planes perpendicular?

When their normal vectors are perpendicular, i.e. \(\vec{n_1}\cdot\vec{n_2}=0\).

When is a line parallel to a plane?

When the line's direction vector is perpendicular to the plane's normal: \(\vec{b}\cdot\vec{n}=0\).

How is the angle between two planes found?

From \(\cos\theta=\dfrac{\vec{n_1}\cdot\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\), using their normals.

What are direction ratios of a line?

Any three numbers proportional to the direction cosines; for the line through \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) they are \(x_2-x_1,\,y_2-y_1,\,z_2-z_1\).

What is the shortest distance between skew lines?

The length of the common perpendicular between them; it is zero exactly when the lines intersect.

How is the distance of a point from a plane found?

For plane \(ax+by+cz=d\) and point \((x_0,y_0,z_0)\), distance \(=\dfrac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}\).

Three Dimensional Geometry

Class 12 Mathematics · Mathematics Part II · Chapter 5

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CBSE Class 12 · Mathematics

Integrals: 35 marks across the four papers

Marks booster + cheatsheet: key concepts in 2 lines each
23 marks in the four papers: Vector Algebra
Integral tables with properties P0–P7
3 methods for differential equations
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CBSE Class 12 · Mathematics

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35 marks: Calculus, shared with Part II
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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:

Flashcards & interactive practice