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Question
Forces of magnitudes `5sqrt(2)` and `10sqrt(2)` units acting in the directions `3hat"i" + 4hat"j" + 5hat"k"` and `10hat"i" + 6hat"j" - 8hat"k"` respectively, act on a particle which is displaced from the point with position vector `4hat"i" - 3hat"j" - 2hat"k"` to the point with position vector `6hat"i" + hat"j" - 3hat"k"`. Find the work done by the forces
Solution
`vec"F"_1 = (3hat"i" + 4hat"j" + 5hat"k")/|3hat"i" + 4hat"j" + 5hat"k"|`
= `(3hat"i" + 4hat"j" + 5hat"k")/sqrt(9 + 16 + 25)`
= `(3hat"i" + 4hat"j" + 5hat"k")/sqrt(50)`
= `(3hat"i" + 4hat"j" + 5hat"k")/(5sqrt(2)`
`vec"F"_2 = (10hat"i" + 6hat"j" - 8hat"k")/sqrt(100 + 36 + 64)`
= `(10hat"i" + 6hat"j" - 8hat"k")/sqrt(200)`
= `(10hat"i" + 6hat"j" - 8hat"k")/(2sqrt(2)`
Resultant force `bar"F" = bar"F"_1 + bar"F"_2`
= `5sqrt(2) bar"F"_1 + 10sqrt(2) bar"F"_2`
= `3hat"i" + 4hat"j" + 5hat"k" + 10hat"i" + 5hat"j" - 8hat"k"`
= `13hat"i" + 10hat"j" - 3hat"k"`
`bar"OA" = 4hat"i" - 3hat"j" - 2hat"k"`
`bar"OB" = 6hat"i" + hat"j" - 3hat"k"`
`bar"d" = bar"AB"`
= `bar"OB" - bar"OA"`
= `2hat"i" + 4hat"j" - hat"k"`
`bar"F" * bar"d" = (13hat"i" + 10hat"j" - 3hat"k")* (2hat"i" + 4hat"j" - hat"k")`
= 26 + 40 + 3
= 69 units
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