Three Dimensional Geometry
Basic concepts of three dimensional geometry class 11 chapter 12, Distance formula , section formula, midpoint formula and centroid of triangle
To locate the position of a point in a plane we need two intersecting mutually perpendicular lines in the plane.
These lines are called coordinate axis and the two points in this case are called the coordinates of the point with respect to the axis.
To locate the position of an object in a space we need three mutually perpendicular planes. These planes are called three coordinate planes.
Three numbers representing the three distances of an object from three coordinate planes are called the coordinates of the point on the object.
So any point in space has three coordinates.
Coordinate axis and coordinate planes in three dimensional planes
Coordinate axis in two dimensions divide the plane into four quadrants.
In three dimensions coordinate planes divide the space(three dimension) into eight parts and each part is called octant, as shown in the figure.
Three Mutually Perpendicular Planes
With the help of quadrants we can make the sign convention for octant as follows.
Cartesian Coordinate System
For xaxis and yaxis Sign convention in different quadrant is given as above.
In three dimension we have one more axis that is zaxis.
For first four octant (I to IV) we take the sign of xaxis and yaxis same as it is in quadrant I to IV and the sign of zaxis is taken positive.
In first octant : xaxis is +ve, yaxis is +ve and zaxis is also +ve. So in first octant point is of the form (+, +, +)
In second octant : xaxis is ve, yaxis is +ve and zaxis is +ve. So in second octant point is of the form (, +, +) and so on up to 4th octant.
For last four octant (V to VIII) we again take the sign of xaxis and yaxis as given in quadrant I to IV and sign of zaxis is taken negative.
In 5th octant : xaxis is +ve, yaxis is +ve and zaxis is ve. So in 5th octant point is of the form (+, +, )
In 6th octant : xaxis is ve, yaxis is +ve and zaxis is ve. So in 6th octant point is of the form (, +, ) and so on up to VIII octant.
So sign convention for eight octant is written as
Octant

I

II

III

IV

V

VI

VII

VIII

Sign

(+,+,+)

(,+,+)

(,,+)

(+,,+)

(+,+,)

(,+,)

(,,)

(+,,)

Coordinates of the point on xaxis is (x, 0, 0),
Coordinates of the point on yaxis is (0, y, 0),
Coordinates of the point on zaxis is (0, 0, z),
Coordinate of the point in xyplane is of the form (x, y, 0),
Coordinate of the point in yzplane is of the form (0, y, z),
Coordinate of the point in zxplane is of the form (x, 0, z),
Coordinates of a point in space is of the form (x, y, z)
Distance between two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) is given by \[PQ=\sqrt{(x_{2}x_{1})^{2}+(y_{2}y_{1})^{2}+(z_{2}z_{1})^{2}}\]
Types of Quadrilaterals By using Distance formula
Let ABCD is a quadrilateral. Then with the help of distance formula find the length of sides AB, BC, CD, DA and diagonals AC and BD
i) If opposite sides are equal but diagonals are not equal then quadrilateral is a parallelogram. or AB = CD, BC = DA and AC ≠ BD
ii) If opposite sides are equal and diagonals are also equal then quadrilateral is a Rectangle. or AB = CD, BC = DA and AC = BD
iii) If all sides are equal and diagonals are not equal then quadrilateral is rhombus. or AB = BC = CD = DA and AC ≠ BD
iv) If all sides are equal and diagonals are also equal then quadrilateral is square. or AB = BC = CD = DA and AC = BD
v) If two pair of adjacent sides are equal and diagonals are not equal then quadrilateral is a Kite.
Here two cases are arises
1. If Point P(x, y, z) divide the line through the points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) in ratio m_{1} : m_{2} internally, then coordinates of point P are given by section formula:
\[P(x,y,z)=\left (
\frac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}},
\frac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}},
\frac{m_{1}z_{2}+m_{2}z_{1}}{m_{1}+m_{2}} \right )\]
2. If Point P(x, y, z) divide the line through the points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) in ratio m_{1} : m_{2} Externally, then coordinates of point P are given by section formula:
\[P(x,y,z)=\left (
\frac{m_{1}x_{2}m_{2}x_{1}}{m_{1}m_{2}}, \frac{m_{1}y_{2}m_{2}y_{1}}{m_{1}m_{2}},
\frac{m_{1}z_{2}m_{2}z_{1}}{m_{1}m_{2}} \right )\]
If Point P(x, y, z) is the midpoints of the line joining the points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) then\[P(x,y,z)=\left (
\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2},\frac{z_{1}+z_{2}}{2} \right )\]
Median:
Median is the line segment which join the vertex of the triangle with the midpoint of the opposite side.
Point of concurrence of all the median of the triangle is called its centroid
Centroid of the triangle whose vertices are A(x_{1}, y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}) and C(x_{3}, y_{3}, z_{3}) is given by\[P(x,y,z)=\left (
\frac{x_{1}+x_{2} +x_{3}}{3},\frac{y_{1}+y_{2}+y_{3}}{3}, \frac{z_{1}+z_{2}+z_{3}}{3}
\right )\]Centroid of the triangle divide the median in 2 : 1
Centroid of the given triangle and the triangle obtained by joining the midpoints of the sides of the triangle are same.
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