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Question
A point P with position vector `(- 14hat"i" + 39hat"j" + 28hat"k")/5` divides the line joining A (1, 6, 5) and B in the ratio 3 : 2, then find the point B.
Solution
Let A, B and P have position vectors a, b and p respectively.
Then `bar"a" = - hat"i" + 6hat"j" + 5hat"k"`,
`bar"p" = (- 14hat"i" + 39hat"j" + 28hat"k")/5`
Now, P divides AB internally in the ratio 3 : 2
∴ `bar"p" = (3bar"b" + 2bar"a")/5`
∴ `5bar"p" = 3bar"b" + 2bar"a"`
∴ `3bar"b" = 5bar"p" - 2bar"a"`
∴ `3bar"b" = 5((- 14hat"i" + 39hat"j" + 28hat"k")/5) - 2(- hat"i" + 6hat"j" + 5hat"k")`
`= - 14hat"i" + 39hat"j" + 28hat"k" + 2hat"i" - 12hat"j" - 10hat"k"`
`= - 12hat"i" + 27hat"j" + 18hat"k"`
∴ `bar"b" = - 4hat"i" + 9hat"j" + 6hat"k"`
∴ coordinates of B are (-4, 9, 6).
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