Question

A force $(4\hat{i}+5\hat{j}-3\hat{k})$ newton is acting at a point $\hat{i}+2\hat{j}+3\hat{k}$. The torque acting a point $3\hat{i}-2\hat{j}-3\hat{k}$ is

Answer 1

Torque is basically the cross product of the position vector r( about a point ) and the force f acting at that particular point.

Here the position vector r,

$=$ position at point $(1,2,3)$ with respect to point $(3\hat{i}-2\hat{j}-3\hat{k})$

$=(\hat{i}+2\hat{j}+3\hat{k})-(3\hat{i}-2\hat{j}-3\hat{k})$

$=-2\hat{i}+4\hat{j}+6\hat{k}$

Force $f=4\hat{i}+5\hat{j}-3\hat{k}$

so, torque $\left(T\right)=r\times f$

$=\left[\begin{array}{ccc}i& j& k\\ -2& 4& 6\\ 4& 5& -3\end{array}\right]$

$=i(-12-30)-j(6-24)+k(-10-16)$

$=-42\hat{i}+18\hat{j}-26\hat{k}$

$=42\hat{i}-18\hat{j}+26\hat{k}$

Hence, the answer is $42\hat{i}-18\hat{j}+26\hat{k}.$