Chapter 11 ยท Introduction

Chapter 11 โ€” Three Dimensional Geometry

Two aircraft fly straight paths at different heights. Will the paths ever meet โ€” and if not, how close do they come? This chapter describes lines in space with direction cosines and equations, measures the angle between two lines, and finds the shortest distance between them.

Air-traffic controllers at a busy airport such as Delhi or Mumbai watch dozens of aircraft at once. Over a short stretch, each aircraft flies a straight line in space: it has a starting point, a direction, and it moves steadily along that direction. Two such paths on a radar screen may appear to cross โ€” but one aircraft may be thousands of metres above the other. In three dimensions two straight lines can do something they can never do on a flat page: they can be neither parallel nor intersecting. Such lines are called skew lines, and the natural safety question for them is: how close do the two paths come?

To answer it you need a precise language for lines in space. First, the direction of a line: three numbers โ€” its direction cosines or, more conveniently, its direction ratios โ€” which you can read off from any two points on it. Next, the equation of a line, in vector form $\vec r=\vec a+\lambda\vec b$ and in Cartesian form, where the single parameter $\lambda$ slides a point along the whole line. With these, the dot and cross products of Chapter 10 do the rest: the angle between two lines comes from a dot product, and the shortest distance between two skew lines comes from a cross product followed by a dot product. The same ideas are used to route cables and pipes in buildings, to check that robot arms do not collide, and to keep aircraft safely apart.

3-D coordinates (Class 11)โ†’Vectors (Ch 10)โ†’Direction cosines & ratios of a lineโ†’Vector & Cartesian equations of a lineโ†’Angle between two linesโ†’Skew lines & shortest distance
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • find the direction cosines and direction ratios of a line, including the line through two given points, and use $l^2+m^2+n^2=1$;
  • decide whether three points are collinear using direction ratios;
  • write the vector and Cartesian equations of a line through a point in a given direction, or through two points, and convert between the two forms;
  • find points on a line at a given distance, test whether a point lies on a line, and find the foot of the perpendicular from a point to a line;
  • find the angle between two lines and the conditions for two lines to be perpendicular or parallel;
  • recognise skew lines and find the shortest distance between two skew lines or two parallel lines, and decide whether two lines intersect;
  • answer the Board's 1-, 2-, 3- and 5-mark questions on this chapter and the JEE Main / JEE Advanced problems built on it.
Teaser โ€” How Close Do Two Flight Paths Come?

Two illustrative straight flight paths (coordinates in km, invented for this lab): the red one is level at a height of 2 km, the blue one is climbing. Their shadows on the ground (dashed) cross โ€” but do the paths themselves meet? Move $\lambda$ (or press Play) to slide the point $P$ along the red path; the lab finds the nearest point $Q$ of the blue path and the gap $PQ$. Drag the picture to turn it. Where is the gap smallest, and what is special about $PQ$ there?

$\lambda$โ€”
$P$ on red pathโ€”
nearest $Q$ on blue pathโ€”
gap $PQ$โ€”
smallest gap seenโ€”

Sections in this chapter

11.1

Direction Cosines and Direction Ratios of a Line

The direction of a line as three numbers; the identity $l^2+m^2+n^2=1$; direction ratios from two points; collinear points.

$l=\pm\dfrac{a}{\sqrt{a^2+b^2+c^2}}$
Open section โ†’11.2

Equation of a Line in Space

Vector and Cartesian equations through a point in a given direction and through two points; points at a given distance; the foot of a perpendicular.

$\vec r=\vec a+\lambda\vec b$
Open section โ†’11.3

Angle between Two Lines and Shortest Distance

The angle between two lines from their direction ratios; perpendicular and parallel lines; skew lines and the shortest distance between two lines.

$d=\left|\dfrac{(\vec b_1\times\vec b_2)\cdot(\vec a_2-\vec a_1)}{|\vec b_1\times\vec b_2|}\right|$
Open section โ†’
โœ… Before you start
This chapter builds on Class 11 Mathematics, Ch. 11 โ€” Introduction to Three Dimensional Geometry (coordinate axes, the coordinates of a point in space, the distance formula $PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$) and on Chapter 10 โ€” Vector Algebra of this course (Sections 10.1โ€“10.4): position vectors $\overrightarrow{OP}=x\hat i+y\hat j+z\hat k$, the direction cosines of a vector, the scalar product $\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta$ and the vector product $\vec a\times\vec b$. In Section 11.3 a $3\times3$ determinant appears, so revise how to expand one (Section 4.1). Quick self-test: find the distance between $(1,2,3)$ and $(4,6,15)$, the scalar product $(2\hat i+\hat j-2\hat k)\cdot(\hat i-2\hat j+3\hat k)$, and $\hat i\times\hat j$. (Answers: $13$; $2-2-6=-6$; $\hat k$.)
โ† 10.4 The Vector (Cross) Product11.1 Direction Cosines and Direction Ratios of a Line โ†’