# Find the position vector of point R which divides the line joining the points P and Q whose position vectors are 2i^-j^+3k^ and -5i^+2j^-5k^ in the ratio 3:2(i) internally(ii) externally - Mathematics and Statistics

Sum

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are 2hat"i" - hat"j" + 3hat"k" and -5hat"i" + 2hat"j" - 5hat"k" in the ratio 3:2
(i) internally
(ii) externally

#### Solution

Let bar("p"), bar("q") and bar("r") be the position vectors of points P, Q and R respectively.

∴ bar("p") = 2hat"i" - hat"j" + 3hat"k", bar("q") = -5hat"i" + 2hat"j" - 5hat"k", m:n = 3:2

(i) R divides the line PQ internally in the ratio 3:2

∴ By using section formula for internal division,

bar("r") = ("m"bar("q") + "n"bar("p"))/("m" + "n")

= (3(-5hat"i" + 2hat"j" - 5hat"k") + 2(2hat"i" - hat"j" + 3hat"k"))/(3 + 2)

= (-15hat"i" + 6hat"j" - 15hat"k" + 4hat"i" - 2hat"j" + 6hat"k")/5

∴ bar("r") = (-11hat"i" + 4hat"j" - 9hat"k")/5

= (-11)/5hat"i" + 4/5hat"j" - 9/5hat"k"

(ii) R divides the line PQ externally in ratio 3:2

∴ By using section formula for external division,

bar("r") = ("m"bar("q") - "n"bar("p"))/("m" - "n")

= (3(-5hat"i" + 2hat"j" - 5hat"k") - 2(2hat"i" - hat"j" + 3hat"k"))/(3 - 2)

= (-15hat"i" + 6hat"j" - 15hat"k" - 4hat"i" + 2hat"j" - 6hat"k")/1

∴ bar("r") = -19hat"i" + 8hat"j" - 21hat"k"

Concept: Section Formula
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