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Find the angle between force vec(F) =(3hat(i)+4hat(j)-5hat(k)) unit and displacement vec(d)=(5hat(i)+4hat(j)+3hat(k)) unit , Also find the projection of F on d . |
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Answer» Solution :`VEC(F) =3HAT(i)+4hat(j)-5hat(k)` `vec(d) =5hat(i)+4hat(j)+3hat(k)` `vec(F).vec(d) =(3hat(i)+4hat(j)-5hat(k)).(5hat(i)+4hat(j)+3hat(k))` ` = 15+16 - 15` = 16 unit ` | vec(F) | = sqrt((3)^(2)+(4)^(2)+(-5)^(2))` `= sqrt(9+16+25)` ` =sqrt(50)` `|vec(F)| =sqrt((3)^(2)+(4)^(2)+(-5)^(2))` `=sqrt(25+16+9)` `= sqrt(50)` `|vec(d)|=sqrt((3)^(2)+(4)^(2)+(-5)^(2))` Now `vec(F).vec(d) =| vec(F) | |vec(d)| cos theta ` ` :. cos theta= (vec(F).vec(d))/(|vec(F)||vec(d)|)` `= 16/((sqrt(15))(sqrt(50)))` ` =16/15` ` :. cos theta =0.32` `sin(90^(@) - theta) = 0.32 ` `90^(@) - theta =18^(@)40` ` :. theta = 90^(@) - 18^(@) 40` ` :.theta = 71^(@) 21 ` PROJECTION of F on d =`F cos theta ` ` = sqrt(50) xx0.32 ` ` = 2.263` N 
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