INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

1 Coordinate axes and coordinate planes Let X OX, Y OY, Z OZ be threemutually perpendicular lines that pass through a point O such that X OX and Y OY liesin the plane of the paper and line Z OZ is perpendicular to the plane of paper. Thesethree lines are called rectangular axes ( lines X OX, Y OY and Z OZ are called x-axis,y-axis and z-axis). We call this coordinate systema THREE DIMENSIONAL space, or simply THREE axes taken together in pairs determinexy, yz, zx-plane , , THREE coordinate planes. Eachplane divide the space in two parts and the threecoordinate planes together divide the space intoeight regions (parts) called octant, namely (i) OXYZ(ii) OX YZ (iii) OXY Z (iv) OXYZ (v) OXY Z (vi) OX YZ (vii) OX Y Z (viii) OX Y Z.

2 ( ).Let P be any point in the space, not in a coordinateplane, and through P pass planes parallel to thecoordinate planes yz, zx and xy meeting thecoordinate axes in the points A, B, C planes are(i) ADPF || yz-plane(ii)BDPE || xz-plane(iii)CFPE || xy-planeThese planes determine a rectangular parallelopiped which has THREE pairs of rectangularfaces(A D P F, O B E C),(B D P E, C F A O) and (A O B D, FPEC) (Fig ) Coordinate of a point in space An arbitrary point P in THREE -dimensionalspace is assigned coordinates (x0, y0, z0) provided thatChapter12 INTRODUCTION TO THREEDIMENSIONAL GEOMETRYFig.

3 (1) the plane through P parallel to the yz-plane intersects the x-axis at (x0, 0, 0);(2) the plane through P parallel to the xz-plane intersects the y-axis at (0, y0, 0);(3) the plane through P parallel to the xy-plane intersects the z-axis at (0, 0, z0).The space coordinates (x0, y0, z0) are called the Cartesian coordinates of P or simplythe rectangular coordinates of we can say, the plane ADPF( ) is perpendicular to the x-axis or x-axis is perpendicular to the plane ADPF andhence perpendicular to every line in the , PA is perpendicular to OX and OXis perpendicular to PA.

4 Thus A is the foot ofperpendicular drawn from P on x-axis anddistance of this foot A from O is x-coordinateof point P. Similarly, we call B and C are thefeet of perpendiculars drawn from point P onthe y and z-axis and distances of these feet Band C from O are the y and z coordinates ofthe point the coordinates x, y z of a point P are the perpendicular distance of P from thethree coordinate planes yz, zx and xy, Sign of coordinates of a point The distance measured along or parallel to OX,OY, OZ will be positive and distance moved along or parallel to OX , OY , OZ will benegative.

5 The THREE mutually perpendicular coordinate plane which in turn divide thespace into eight parts and each part is know as octant. The sign of the coordinates ofa point depend upon the octant in which it lies. In first octant all the coordinates arepositive and in seventh octant all coordinates are negative. In third octant x, y coordinatesare negative and z is positive. In fifth octant x, y are positive and z is negative. In fourthoctant x, z are positive and y is negative. In sixth octant x, z are negative y is the second octant x is negative and y and z are IIIIIIIVVVIVIIVIIIC oordinatesOXYZOX YZ OX Y Z OXY Z OXYZ OX YZ OX Y Z OXY Z x+ ++ +y++ ++ z++++ INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 209 Fig.

6 EXEMPLAR PROBLEMS Distance formula The distance between two points P (x1, y1, z1) and Q (x2, y2,z2) is given by222212 121PQ)()()xxyy zz= + + A paralleopiped is formed by planes drawn through the points (x1, y1, z1) and (x2, y2, z2)parallel to the coordinate planes. The length of edges are x2 x1, y2 y1, z2 z1 andlength of diagonal is 222212 121()( )()xxyyzz + + . Section formula The coordinates of the point R which divides the line segmentjoining two points P(x1, y1, z1) and Q(x2, y2, z2) internally or externally in the ratio m : nare given by 21 21 2121 21 21,,,, ,,mxnxmynymznzmxnxmynymznzmnmnmnmnmnmn +++ +++ , coordinates of the mid-point of the line segment joining two points P (x1, y1, z1) andQ (x2, y2, z2) are 121 212,,222xxy yzz+++.

7 The coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1), (x2, y2, z2)and x3, y3, z3 are 1231 2 3123,,333xxxyy yzzz++ ++ ++ . Solved ExamplesShort Answer TypeExample 1 Locate the points (i) (2, 3, 4)(ii) ( 2, 2, 3) in (i) To locate the point (2, 3, 4) in space, wemove 2 units from O along the positivedirection of x-axis. Let this point be A(2, 0, 0). From the point A moves 3 unitsparallel to +ve direction of point be B (2, 3, 0). From the pointB moves 4 units along positive directionof z-axis.

8 Let this point be P (2, 3, 4)Fig.( ).Fig. TO THREE DIMENSIONAL GEOMETRY 211 (ii)From the origin, move 2 units along the negative direction of x-axis. Let this pointbe A ( 2, 0, 0). From the point A move 2 units parallel to negative direction this point be B ( 2, 2, 0). From B move 3 units parallel to positive direction ofz - axis. This is our required point Q ( 2, 2, 3) ( )Fig. 2 Sketch the plane (i) x = 1 (ii) y = 3 (iii) z = 4 Solution(i) The equation of the plan x = 0 represents the yz-plane and equation of the planex = 1 represents the plane parallel to yz-plane at a distance 1 unit above yz-plane.

9 Now, we draw a plane parallel to yz- plane at a distance 1 unit above yz-plane (a).(ii)The equation of the plane y = 0 represents the xz plane and the equation of theplane y = 3 represents the plane parallel to xz plane at a distance 3 unit above xzplane (Fig. (b)).(iii)The equation of the plane z = 0 represents the xy-plane and z = 3 represents theplane parallel to xy-plane at a distance 3 unit above xy-plane (Fig. (c)).(a)(b)(c)Fig. EXEMPLAR PROBLEMS MATHEMATICSE xample 3 Let L, M, N be the feet of the perpendiculars drawn from a point P (3, 4, 5)on the x, y and z-axes respectively.

10 Find the coordinates of L, M and Since L is the foot of perpendicular from P on the x-axis, its y and z co-ordinates are zero. The coordinates of L is (3, 0, 0). Similarly, the coordinates of M andN are (0, 4, 0) and (0, 0, 5), 4 Let L, M, N be the feet of theperpendicular segments drawn from a pointP (3, 4, 5) on the xy, yz and zx-planes, are the coordinates of L, M and N?Solution Since L is the foot of perpendicularsegment from P on the xy-plane, z-coordinate iszero in the xy-plane.