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Question
The position vector of the point which divides the join of points `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1 is ______.
Options
`(3vec"a" - 2vec"b")/2`
`(7vec"a" - 8vec"b")/4`
`(3vec"a")/4`
`(5vec"a")/4`
Solution
The position vector of the point which divides the join of points `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1 is `(5vec"a")/4`.
Explanation:
The given vectors are `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1.
∴ The position vector of the required point c which divides the join of the given vectors `vec"a"` and `vec"b"` is
`vec"c" = ("m"_1x_2 + "m"_2x_1)/("m"_1 + "m"_2)`
= `(1 * (2vec"a" - 3vec"b") + 3(vec"a" + vec"b))/(3 + 1)`
= `(2vec"a" - 3vec"b" + 3vec"a" + 3vec"b")/4`
= `(5vec"a")/4`
= `5/4 vec"a"`
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