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# Basic Concepts of Three Dimensional JEE Notes | EduRev

## JEE : Basic Concepts of Three Dimensional JEE Notes | EduRev

The document Basic Concepts of Three Dimensional JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 11.
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Distance Between Two Points
Let P and Q be two given points in space. Let the co-ordinates of the points P and Q be  and  with respect to a set OX, OY, OZ of rectangular axes.

The position vectors of the points P and Q are given by

Now we have

B. Section Formula

(for external division take â€“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

or    [Using (1), (2) and (3)]

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 Î”ABC. Thus D is the mid point of BC.

Now if G is the centroid of Î”ABC, then G divides AD in the ratio 2 : 1. Let the co-ordinates of G be (x, y, z).

Centroid Of A Tetrahedron :

Let ABCD be a tetrahedron, the co-ordinates of whose vertices are  = 1, 2, 3, 4.

Let  be the centroid of the face ABC of the tetrahedron. Then the co-ordinates of  are

The fourth vertex D of the tetrahedron does not lie in the plane of Î”ABC. 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

Ex.1 P is a variable point and the co-ordinates of two points A and B are (â€“2, 2, 3) and (13, â€“3, 13) respectively. Find the locus of P if 3PA = 2PB.

Sol. Let the co-ordinates of P be (x, y, z).

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

9{(x + 2)2 + (y â€“ 2)2 + (z â€“ 3)2} = 4 {(x â€“ 13)2 + (y + 3)2 + (z â€“ 13)2}

or 9 {x2 + y2 + z2 + 4x â€“ 4y â€“ 6z + 17} = 4{x2 + y2 + z2 â€“ 26x + 6y â€“ 26z + 347}

or 5x2 + 5y2 + 5z2 + 140x â€“ 60 y + 50 z â€“ 1235 = 0

or x2 + y2 + z2 + 28x â€“ 12y + 10z â€“ 247 = 0

This is the required locus of P.

Ex.2 Find the ratio in which the xy-plane divides the join of (â€“3, 4, â€“8) and (5, â€“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 (â€“3,4, â€“8) and (5, â€“6, 4) in the ratio ,  in the point R. Therefore, the co-ordinates of the point R are

....(1)

But on xy-plane, the z co-ordinate of R is zero

(4Î¼ - 8) / (Î¼ + 1) = 0, or Î¼ = 2. Hence  Î¼ : 1 = 2 : 1. Thus the required ratio is 2 : 1.
Again putting  the co-ordinates of the point R become (7/3, â€“8/3, 0).

Ex.3 ABCD is a square of side length â€˜aâ€™. 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 coordinates of the point A are (0, a sin Î¸) and that of C are (a cos Î¸ + a sin Î¸, a cos Î¸)

...(2)

Form (1) and (2),

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## Mathematics (Maths) Class 11

158 videos|179 docs|161 tests

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