The moment about a line through the origin having the direction of `1 12hati -4hatj -3hatk` is

Options

A. `760 / 13`
B. ` (-760)/13`
C. ` 76/13`
D. ` (760)/3`

Answer

B

Explanation

Let `vecF` be the force, then
` vecF = (30 (12hati -4hatj -3hatk))/(sqrt(12^(2) + (-4)^(2) + ( -3)^(2) ))= 30/13 (12hati -4hatj -3hatk)`
Suppose the force ` vecF` acts at point P ( -4,2,5) .The moment of ` vecF` acting at P about a line in the direction ` 2hati - 2hatj +hatk` is equal to the resolve part along the line of the moment fo ` vecF ` about a point on the line. It is given that the line passes through the origin O . So we choose O as a point on the line . Let ` vec(OP) = vecr` then ` vecr = vec(OP)` = Position vector of P - position vector of O
` Rightarrow vecr = ( -4hati +2hatj + 5hatk) -vec0 = - 4hati + 2hatj + 5hatk`

Let `vecM` ve the moment of `vecF` about O. Then
`vecM = vecr xx vecF`
` Rightarrow vecM = (-4hati +2hatj +5hatk) xx 30/13 (12 hati - 4hati -3hatk)`
`vecM=30/13|{:(hati,hatj,hatk),(-4,2,5),(12,-4,-3):}|= 30/13 (14hati+48hatj -8hatk)`
Let `veca` be unit vector in the direction of `2hati -2hatj +hatk` , then
Thus, the moment of `vecF` about the given line
`veca= (2hati-2hatj+hatk)/(sqrt(4+4+1))=1/3 (2hati-2hatj+hatk)`
`=vecM.veca=30/13(14hati+48hatj-8hatk).""1/3(2hati-2hatj +hatk)= (-760)/13`