Derive the coordinates of the points $R (x, y, z)$ dividing the line joining the points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ internally in the ratio $m : n$ .

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Solution

see the above diagram

triangle ASP is simiar to triangle BTP

$\Rightarrow \frac{m}{n} = \frac{A P}{B P} = \frac{A S}{B T} = \frac{S L - A L}{B M - T M} = \frac{N P - A L}{B M - P N} = \frac{z - z_{1}}{z_{2} - z}$

$\Rightarrow z = \frac{m z_{2} + n z_{1}}{m + n}$

similarly we can prove for x and y

$\Rightarrow x = \frac{m x_{2} + n x_{1}}{m + n}$

$\Rightarrow y = \frac{m y_{2} + n y_{1}}{m + n}$

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Similar questions

Let $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ be two points and let R(x, y, z) be a point on PQ dividing it in the ratio m:n. Prove that
$x = \frac{m x_{2} + n x_{1}}{m + n}, y = \frac{m y_{2} + n y_{1}}{m + n}$ and $z = \frac{m z_{2} + n z_{1}}{m + n}$ .

The coordinates of the mid-point of the line segment joining two points P(X₁, y₁, z₁) and Q(x₂, y₂, z₂) are:

Q. Find the scalar components and magnitude of the vector joining the points

P (x_{1}, y_{1}, z_{1})

and

Q (x_{2}, y_{2}, z_{2})

.
.

Find the scalar components and hence magnitude of the vector joining the points $P (x_{1}, y_{1}, z_{1}) a n d Q (x_{2}, y_{2}, z_{2})$ .

Which of the follwing statements are correct ?
1. The coordinates of the point R which divides the line

segment joining two points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$

externally in the ratio m:n are $(\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n}, \frac{m z_{2} + n z_{1}}{m + n})$ .

2. If R divides PQ internally in the ratio m:n, then its coordinates

are obtained by replacing n by -n in the statement 1.

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Derive the coordinates of the points R(x,y,z) dividing the line joining the points P(x1,y1,z1) and Q(x2,y2,z2) internally in the ratio m:n.

see the above diagramtriangle ASP is simiar to triangle BTP⇒mn=APBP=ASBT=SL−ALBM−TM=NP−ALBM−PN=z−z1z2−z⇒z=mz2+nz1m+nsimilarly we can prove for x and y⇒x=mx2+nx1m+n⇒y=my2+ny1m+n

Derive the coordinates of the points $R (x, y, z)$ dividing the line joining the points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ internally in the ratio $m : n$ .