for internal and external division
Let P and Q be two points represented by the position vectors `vec (OP)` and `vec (OQ)`, respectively, with respect to the origin O. Then the line segment joining the points P and Q may be divided by a third point, say R, in two ways – internally and externally. Then we find the position vector `vec (OR)` for the point R with respect to the origin O . We take the two cases one by one.
Case I - Fig.
When R divides PQ internally . If R divides `vec (PQ)` such that m `vec (RQ)` = n `vec (PR)`, where m and n are positive scalars, we say that the point R divides `vec (PQ)` internally in the ratio of m : n. Now from triangles ORQ and OPR, we have
`vec (RQ) = vec (OQ) - vec(OR) = vec b - vec r` and
`vec (PR) = vec (OR) - vec (OP) = vec r - vec a`, Therefore we have
`m(vec b - vec r) = n (vec r - vec a),`
`vec r = (m vec b + n vec a)/(m+n)` ( On simplification)
Hence, the position vector of the point R which divides P and Q internally in the ratio of m : n is given by
`vec (OR) = (m vec b + n vec a)/(m + n)`
Case II - When R divides PQ externally Fig.
The position vector of the point R which divides the line segment PQ externally in the ratio m : n i.e. `(PR)/(QR) = m/n` is given by
`vec (OR) = (m vec b - n vec a)/(m - n)`
Remark: If R is the midpoint of PQ , then m = n. And therefore, from Case I, the midpoint R of `vec (PQ)`, will have its position vector as
`vec (OR) = (vec a + vec b)/2`
Shaalaa.com | Section Formula for Internal
If `bar p = hat i - 2 hat j + hat k and bar q = hat i + 4 hat j + 2 hat k` are position vector (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2:1
If point C `(barc)` divides the segment joining the points A(`bara`) and B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`
If `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that
`bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `