Introduction to Three Dimensional Geometry

Chapter 12

Introduction to Three Dimensional Geometry

x- axis : (x,0,0)

y- axis : (0,y,0)

z- axis : (0,0,z)

XY-plane : (x,y,0)

YZ-plane : (0,y,z)

ZX-plane : (x,0,z)

+ + + : I

– + + : II

– –  + : III

+ – + :  IV

+ + – : V

– + – : VI

– – – : VII

+ – –  : VIII

Q. (4,-2,3)

IV   XOYIZ

Q. (-4,2,5)

II   XIOYZ

The distance of the point (a,b,c) from x-axis

The distance of the point (a,b,c) from y-axis

The distance of the point (a,b,c) from z-axis

Q. Find the distance of  the point (2,3,4) from y-axis

The image of (a,b,c) w.r.to XY-plane  : (a,b,-c)

The image of (a,b,c) w.r.to YZ-plane  : (-a,b,c)

The image of (a,b,c) w.r.to ZX-plane  : (a,-b,c)

Q. Find the image of (2,-3,-4) w.r.to XY-plane

(2,-3,4)

Distance formula

Q. Using distance formula show that the points P(-2,3,5), Q(1,2,3) and R(7,0,-1) are collinear

 

Points are collinear

Q. Find the equation of set of points P which are equidistant from A(1,2,3) and B(3,2,-1)

Let the coordinates of the point P be (x,y,z)

Section Formula

Internal division

External division

Midpoint =

Q. Find the ratio in which the line segment joining (4,8,10) and (6,10,-8) is divided by YZ- plane

Let k:1 be the ratio

x coordinate of a point in YZ-plane is 0

k.6+ 1.4 = 0

6k+4 = 0

Externally in the ration 2 : 3

Q. Find the coordinates of a point which trisect the line segment joining (4,2,-6) and

 (10,-16,6)

A divides the line segment in the ratio 1:2

B divides the line segment in the ratio 2:1

Q. Using section formula prove that the three points A(-4,6,10), B(14,0,-2) and C(2,4,6) are collinear

Assume that B divides AC in the ratio k:1

2k-4 = 14(k+1)

2k-4 = 14k + 14

-12k = 18

K = –  3/2

B divides AC in the ratio 3:2 externally

The points are collinear

Centroid of a triangle

The meeting point of three medians of a triangle is called centroid. Centroid divides each median in thr ratio 2:1

Centroid =

Q. If the origin is the centroid of a triangle with vertices (2a,2,6), (-4,3b,-10) and (8,14,2c) find the values of a,b and c

Centroid

2a+4 = 0

a = -2

3b+16 = 0

b = – 16/3

(2c-4)/3 = 0

2c – 4 = 0

c = 2

 

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