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Question
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"`
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