Question

Derive an expression for the coordinates of a point that divides the line joining the points A(x₁, y₁, z₁,) and B (x₂, y₂, z₂.) internally in the ratio m : n. Hence, find the coordinates of the midpoint of AB where A = (1, 2, 3) and B = (5, 6, 7). 

Answer 1

Let A(x₁, y₁, z₁) and B(x₂, y₂, z₂) be the given points

Let R(x, y, z) divide PQ internally in the ratio m : n

Draw the line segments AP, BQ, CR perpendicular to xy-plane.

∴ AP ∥ BQ ∥ CR

∴ AP, BQ, CR lines are lie in one plane.

So the points P, Q and R lie in a straight line.

And the points intersects the plane and xy plane.  

Through the point R draw a parallel line AB to the line segment PQ.

The line AB intersects the line segment LP externally at the point A and the line segment MQ at the point B.

From the figure we have that the triangles ALN and RBQ are similar triangles

So we can write,

\(\frac{AL}{MB} = \frac{LN}{NB} = \frac mn\)

\(\frac{AP - PL}{BQ - MQ} = \frac mn\)

\(\frac {NR - LP}{QB - NR} = \frac mn\)

\(\frac mn = \frac{z - z_1}{z_2 - z} = \frac mn\)

\(n(z - z_1) = m(z_2 - z)\)

\(nz - nz_1 = mz_2 - mz\)

\(nz + mz = nz_1 + mz_2\)

\(z(n + m) = nz_1 + mz_2\)

\(z(m + n) = mz_2 + nz_1\)

\(z(m + n) = mz_2 + nz_1\)

\(z = \frac{mz_2 + nz_1}{m + n}\)

Similarly we can find x and y

Therefore \(x = \frac{mx_2 + nx_1}{m +n} \) and \(x = \frac{my^2 + ny_1}{m + n}\)

Therefore the expression for the coordinate of the point that divides the line joining A and B in the ratio m:n is

\(N(x, y, z) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m +n}, \frac{mz_2 + nz_1}{m + n}\right)\)

Answer 2

Let the two given points be P (x₁, y₁, z₁,) and Q (x₂, y₂, z₂). 

Let the point R (x, y, z) divide PQ in the given ratio m : n internally, Draw PL, 

QM and RN perpendicular to the XY-plane. 

Obviously PL ∥ RN ∥ QM and feet of these perpendiculars lie in a XY-plane. 

The points L, M and N will lie on a line which is the intersection of the plane containing PL, 

RN and QM with the XY-Plane. 

Through the point R draw a line ST parallel to the line LM. 

Line ST will intersect the line LP externally at the point S and the line MQ at T, as shown in Fig. 

Also note that quadrilaterals LNRS and NMTR are parallelograms. 

The triangles PSR and QTR are similar. Therefore,

m/n = PR/QR = SP/QT = (SL - PL)/(QM - TM) = (NR - PL)/(QM - NR) = (Z - z₁)/(Z₂ - z)

This implies z = (mz₂ + nz₁)/(m + n)

Similarly, by drawing perpendiculars to the XZ and YZ-planes, we get

y = (my₂ + ny₁)/(m + n) and x = (mx₂ + nx₁)/(m + n)

Hence, the coordinates of the point R which divides the line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m: n are

R(x, y, z) = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)).