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 \(=\dfrac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}\).

CBSE · Senior secondary

CBSE Class 12 — Notes, Chapters & Practice Quizzes

Master every Class 12 chapter — the base your board marks and your NEET / JEE / CUET prep both stand on.

13 subjects 166 chapters 171 practice quizzes NCERT syllabus

56 quiz attempts so far Dual AI-verified questions 10 quizzes free, then 10 credits

Notice Board

Students preparing for CBSE Class 12 Board Examination

View all

Formula of the day

Physics · EM Induction · 13 formulae in the cheatsheet

Trap of the day

Lens/mirror sign convention: distances against the incident light are negative. Drop a sign and real becomes virtual.

Chapter 5: Three Dimensional Geometry — Class 12 Mathematics

Mathematics · 13 chapters

Summary, key terms, important questions and a practice quiz with AI diagnosis for each.

Chapter 5: Three Dimensional Geometry

Try one from this chapter

The vector equation of a line is:

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.

Important questions

Explore interactively

Key-term flashcards

Flip cards · mark known · keyboard ← → and Space

6 cards

Term1 / 6

Direction ratios

Tap to reveal

Meaning1 / 6

Numbers proportional to the direction cosines of a line.

Tap to flip back

Tap card to flip

Keyboard: ← → to move · Space to flip

Practice quiz · Three Dimensional Geometry

Score on this chapter, climb the leaderboard, and get an AI diagnosis of your mistakes.

Dual AI-verified questions Real exam pattern 2 quizzes free, then 10 credits per quiz

Three Dimensional Geometry

Maths 10 Qs · ~10 min