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In two-dimensional geometry, Section Formula is the concept applied to find the coordinates of a point that divides a line segment internally in a particular ratio. Let us understand the applications of Section Formula to find the coordinates of a particular point that divides a line internally in three-dimensional space.
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Section Formula in Three-Dimension Geometry
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The concept of Sectional Formula is used to find the position of a particular point in space. In order to do so, a coordinate system is required. In three-dimensional space, in order to give the coordinates of a particular point, say P, a fixed coordinate system must be chosen. Thus, by knowing the coordinates (x, y, z) of the point P, we can locate its position in space with ease. Thus, determining the coordinates of a point in space is the main application of Section Formula in Three-Dimensional Geometry.
For a better understanding of the concept, let us look at an example. Consider any two points A (x1, y1, z1) and B (x2, y2, z2) in an XYZ plane. Consider another point P (x, y, z) which divides the line joining the points A and B in the ratio, m:n.
In order to determine the coordinates of this point P, you have to follow the following steps.
- Draw lines AL, PN and BN perpendicular to the plane XY. This results in-
AL || PN || BM
- Three points namely L, M, and N can be formed which lie on a straight line because of the intersection of a plane that contains AL, PN, BM, and also the X-Y plane.
- Then draw a line segment ST that is parallel to LM from the point P.
- You can draw lines in such a way that ST externally intersects AL at S and it internally intersects BM at T.
Thus, as ST is parallel to LM and AL || PN || BM, we can identify the quadrilaterals LNPS and NMTP as parallelograms. The triangles ΔBTP and ΔASP are also identified as similar triangles.
Thus, since ΔASP~Δ?BTP,
From this, we can get
You can follow the above steps again by drawing perpendiculars to XZ and YZ planes. That way, you can get the x and y coordinates of the point P which internally divides the line AB in the ratio m:n. Thus, you will get the equations:
Section Formula (Internally)
The coordinates(x,y,z) of a point P which divides the line segment that joins the points A(x1,y1,z1) and B(x2,y2,z2) in m:n ratio internally is thus given by,
Section Formula (Externally)
The coordinates(x,y,z) of a point P which divides the line segment that joins the points A (x1,y1,z1) and B (x2,y2,z2) in m:n ratio externally is
Thus, when the point P divides the line AB externally in the ratio m:n in a three-dimensional plane, it becomes ‘-n’ instead of ‘n’ in the coordinates (when the point divides the line internally).
Section Formula (Midpoint)
If the point P (x,y,z) happens to be in the middle of the line segments joining the points A (x1,y1,z1) and B (x2,y2,z2) dividing it internally, then the ratio m:n becomes 1:1. Thus, m=n=1. This will make the coordinates be,
Section Formula (Special Case)
If the point P(x,y,z) divides any line segment joining points A(x1,y1,z1) and B(x2,y2,z2) internally in k:1 ratio, then the coordinates will become.
The video below explains this:
Coordinate Geometry Detailed Video Explanation:
Sample Questions
Question: Find out the coordinates of a point P that divides the line segment which joins the points (1, -2, 3) and (3, 4, -5) in the ratio 2:3 both: (4 marks)
- internally
- externally.
Solution:
- Internally :- Let us assume that P (x, y, z) is the point that divides the line segment which joins the points A (1, -2, 3) and B (3, 4, -5) internally in the ratio 2:3. We know that the coordinates of P will be,
- Externally:- Let us assume that P (x, y, z) is the point that divides the line segment which joins the points A (1, -2, 3) and B (3, 4, -5) externally in the ratio 2:3. We know that the coordinates of P will be,
In which m=2, n=3, x1=1, x2=3, y1=-2, y2=4, z1=3, z2=-5
Therefore,
Thus, we get the coordinates of P
(x,y,z)=-3,-14, 19
Question: Prove using Section Formula that the three points (-4, 6, 10), (2, 4, 6) and (14, 0, -2) are collinear. (5 marks)
Solution: Let us assume that the three points are P (-4, 6, 10), Q (2, 4, 6) and R (14, 0, -2). We can say that a point, say S, divides the line segment PQ in the ratio k:1.
Therefore, the coordinates of P will be,
Now to check whether the points P, Q and R coincide with S. So let us check if some of the values of k, by equating the coordinates of the point S and R.
2k-4=14k+1
2k-4=14k+14
2k-14k=14+4
-12k=18
Thus, by equating the equation, (2k-4)/(k+1) = 14, we get a value of k= -3/2
Now let us try applying the value of k into the equation (4k+6)/(k+1). The answer should come as 0 which is the y-coordinate of the point C.
Hence the point of S that divides the line segment joining the points P (-4, 6, 10) and Q (2, 4, 6) externally in the ratio 3:2, is the same as the point R (14, 0, -2).
Thus, it is proved that the points P, Q and R are collinear.
Question: By using the concept of section formula in three-dimensional geometry, find the coordinates of the point R that divides the line segment joining the points P (3, 5, 2) and Q (2, 4, 3) internally in the ratio 4:6. (3 marks)
Solution: The coordinates of the point R (x, y, z) which divides a line segment joining two points P (3, 5, 2) and Q (2, 4, 3) internally in the ratio 4:6 can be found by the formula,
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