Sectional Folmula - Three Dimensional Geometry (3-D), Class 11, Mathematics PDF Download

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B. SECTION FORMULA


(for external division take –ve sign)
To determine the co-ordinates of a point R which divides the joining of two points P and
internally in the ratio m1 : m2. Let OX, OY, OZ be a set of rectangular axes.
The position vectors of the two given points are given by


Also if the co-ordinates of the point R are (x, y, z), then
Now the point R divides the join of P and Q in the ratio m1 : m2, so that

Remark : The middle point of the segment PQ is obtained by putting m1 = m2. Hence the
co-ordinates of the middle point of PQ are  

CENTROID OF A TRIANGLE :
Let ABC be a triangle. Let the co-ordinates of the vertices A, B and C be
and respectively. Let AD be a median of the ΔABC. Thus D is the mid point of BC.

Now if G is the centroid of ΔABC, then G divides AD in the ratio 2 : 1. Let the co-ordinates of G be

CENTROID OF A TETRAHEDRON :
Let ABCD be a tetrahedron, the co-ordinates of whose vertices are (x_r, y_r, z_r), r = 1, 2, 3, 4.
Let G₁ be the centroid of the face ABC of the tetrahedron. Then the co-ordinates of G₁ are

The fourth vertex D of the tetrahedron does not lie in the plane of DABC. We know from statics that the centroid of the tetrahedron divides the line DG1 in the ratio 3 : 1. Let G be the centroid of the tetrahedron and if (x, y, z) are its co-ordinates, then

 Similarly 

Ex.1 P is a variable point and the co-ordinates of two points A and B are (–2, 2, 3) and (13, –3, 13) respectively. Find the locus of P if 3PA = 2PB.
Sol.Let the co-ordinates of P be (x, y, z).

....(1) and 

  ....(2)

Now it is given that 3PA = 2PB i.e., 9PA² = 4PB². ....(3)

Putting the values of PA and PB from (1) and (2) in (3), we get

 or
or 

This is the required locus of P.

Ex.2 Find the ratio in which the xy-plane divides the join of (–3, 4, –8) and (5, –6, 4). Also find the point of intersection of the line with the plane.
Sol. Let the xy-plane (i.e., z = 0 plane) divide the line joining the points (–3,4, –8) and (5, –6, 4) in the ratio μ : 1, in the point R. Therefore, the co-ordinates of the point R are

But on xy-plane, the z co-ordinate of R is zero ∴​  (4μ -  8) / (μ + 1) = 0, or μ = 2. Hence  μ : 1 = 2 : 1. Thus the required ratio is 2 : 1.
Again putting μ = 2 in (1), the co-ordinates of the point R become (7/3, -8/3, 0).

Ex.3 ABCD is a square of side length ‘a’. Its side AB slides between x and y-axes in first quadrant. Find the locus of the foot of perpendicular dropped from the point E on the diagonal AC, where E is the midpoint of the side AD.
Sol. Let vertex A slides on y-axis and vertex B slides on x-axis coordi nates of the point A are (0, a sin q) and that of C are (a cos q + a sin q, a cos q)

​

   ...(2)

Form (1) and (2),

⇒  is the locus of the point F..