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Question
The position vectors of three consecutive vertices of a parallelogram are `hat"i" + hat"j" + hat"k", hat"i" + 3hat"j" + 5hat"k"` and `7hat"i" + 9hat"j" + 11hat"k"`. Find the position vector of the fourth vertex.
Solution
Let, `bar"a" = hat"i" + hat"j" + hat"k"`
`bar"b" = hat"i" + 3hat"j" + 5hat"k"`
`bar"c" = 7hat"i" + 9hat"j" + 11hat"k"`
and `bar"d"` be the position vectors of four vertices of parallelogram ABCD.
∴ `bar"AB" = bar"DC"` ....[Opposite sides of parallelogram]
∴ `bar"b" - bar"a" = bar"c" - bar"d"`
∴ `bar"d" = bar"a" + bar"c" - bar "b"`
∴ `bar"d" = (hat"i" + hat"j" + hat"k") + (7hat"i" + 9hat"j" + 11hat"k") - (hat"i" + 3hat"j" + 5hat"k")`
= `(1 + 7 - 1)hat"i" + (1 + 9 - 3)hat"j" - (1 + 11 - 5)hat"k"`
= `7hat"i" + 7hat"j" + 7hat"k"`
= `7(hat"i" + hat"j" + hat"k")`
∴ Position vector of the fourth vertex of the parallelogram is `7(hat"i" + hat"j" + hat"k")`.
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