Question

Find the magnitude and direction cosines of the torque of a force represented by `3hat"i" + 4hat"j" - 5hat"k"` about the point with position vector `2hat"i" - 3hat"j" + 4hat"k"`  acting through a point whose position vector is `4hat"i" + 2hat"j" - 3hat"k"`

Answer 1

`bar"OA" = 2hat"i" - 3hat"j" + 4hat"k"`

`bar"OB" = 4hat"i" + 2hat"j" - 3hat"k"`

`hat"r" = bar"AB" = bar"OB" - bar"OA"`

= `2hat"i" + 5hat"j" - 7hat"k"`

= `3hat"i" + 4hat"j" - 5hat"k"`

Torque `bar"M" xx bar"r" xx bar"F" = |(hat"i", hat"j", hat"k"),(2, 5, -7),(3, 4, -5)|`

= `hat"i"(- 25 + 28) - hat"j"(- 10 + 21) + hat"k"(8 - 15)`

= `3hat"i" - 11hat"j" - 7hat"k"`

Troque = `|bar"r" xx bar"F"|`

= `sqrt(9 + 121 + 49(`

= `sqrt(179)`

Direction cosine's `[3/sqrt(179), (-11)/sqrt(179), (-7)/sqrt(179)]`