**MATHS :: Lecture 01 :: 3D Analytical Geometry**

__Three dimensional Analytical geometry__

Let OX ,OY & OZ be mutually perpendicular straight lines meeting at a point O. The extension of these lines OX1, OY1 and OZ1 divide the space at O into octants(eight). Here mutually perpendicular lines are called X, Y and Z co-ordinates axes and O is the origin. The point P (x, y, z) lies in space where x, y and z are called x, y and z coordinates respectively

.

where NR = x coordinate, MN = y coordinate and PN = z coordinate

**Distance between two points**** **

The distance between two points A(x1,y1,z1) and B(x2,y2,z2) is

dist AB =

In particular the distance between the origin O (0,0,0) and a point P(x,y,z) is

OP =

**The internal and External section**

Suppose P(x1,y1,z1) and Q(x2,y2,z2) are two points in three dimensions.

P(x1,y1,z1) A(x, y, z) Q(x2,y2,z2)

The point A(x, y, z) that divides distance PQ internally in the ratio m1:m2 is given by

A = |

Similarly

P(x1,y1,z1) and Q(x2,y2,z2) are two points in three dimensions.

P(x1,y1,z1) Q(x2,y2,z2) A(x, y, z)

The point A(x, y, z) that divides distance PQ externally in the ratio m1:m2 is given by

A = |

If A(x, y, z) is the midpoint then the ratio is 1:1

A = |

**Problem**

Find the distance between the points P(1,2-1) & Q(3,2,1)

PQ= ===2

**Direction Cosines**

Let P(x, y, z) be any point and OP = r. Let a,b,g be the angle made by line OP with OX, OY & OZ. Then a,b,g are called the direction angles of the line OP. cos a, cos b, cos g are called the direction cosines (or dc’s) of the line OP and are denoted by the symbols I, m ,n

.**Result**

By projecting OP on OY, PM is perpendicular to y axis and the also OM = y

Similarly,

(i.e) *l = m = n = *

\*l2 + m2 + n2* =

(Distance from the origin)

\ l2 + m2 + n2 =

l2 + m2 + n2 = 1

(or) cos2a + cos2b + cos2g = 1.

__Note__** :-**

The direction cosines of the x axis are (1,0,0)

The direction cosines of the y axis are (0,1,0)

The direction cosines of the z axis are (0,0,1)

__Direction ratios__

Any quantities, which are proportional to the direction cosines of a line, are called direction ratios of that line. Direction ratios are denoted by a, b, c.

If l, m, n are direction cosines an a, b, c are direction ratios then

*a **µ** l, b **µ** m, c **µ** n*

(ie*) a = kl, b = km, c = kn*

(ie) (Constant)

(or) (Constant)

__To find direction cosines if direction ratios are given__

If a, b, c are the direction ratios then direction cosines are

*l = *

* similarly m = *(1)

* n = *

*l2+m2+n2 = *

(ie) 1 =

Taking square root on both sides

K =

\

**Problem**

1. Find the direction cosines of the line joining the point (2,3,6) & the origin.

**Solution **

By the distance formula

**Direction Cosines are r
**l = cos µ =

o

n = cos g =

2. Direction ratios of a line are 3,4,12. Find direction cosines

**Solution**

Direction ratios are 3,4,12

(ie) a = 3, b = 4, c = 12

Direction cosines are

*l*=

*m*=

*n*=

**Note**

- The direction ratios of the line joining the two points A(x1, y1, z1) &

B (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1) - The direction cosines of the line joining two points A (x1, y1, z1) &

B (x2, y2, z2) are

r = distance between AB.

Download this lecture as PDF here |