Forces $F_1$, $F_2$, $F_3$ and $F_4$ act on a point on a body with the following vectors: $$ \begin{align} F_1 &= 2i - j + k, & r_1 &= 3i + j + 5k \\ F_2 &= 8i - 6j + 6k, & r_2 &= 5i - 2j -k \\ F_3 &= 4i + 3k & r_3 &= i - 6j = 7k\\ F_4 &= i - 5j & r_4 &= i + 6k \end{align} $$ Show that the body is in equilibrium, that is:
$$(r_1 \times F_1) + (r_2 \times F_2) + (r_3 \times F_3) + (r_4 \times F_4) = 0$$
So far I have this:
$$ F_1 = 2i - j + k \qquad r_1 = 3i +j + 5k $$
$$\begin{align} F_1 \times r_1 = \begin{vmatrix} i & j & k\\ - & F_1 & -\\ - & r_1 & - \\ \end {vmatrix} &=\left( \begin{vmatrix} -1 & 1\\ 1 & 5\\ \end{vmatrix},\ -\begin{vmatrix} 2 & 1\\ 3 & 5\\ \end{vmatrix},\ \begin{vmatrix} 2 & -1\\ 3 & 1\\ \end{vmatrix} \right)\\ &= (5 -1)i -(10 - 3)j + (2-(-3))k \\ &= 6i - 7j + 5k \end{align} $$
So far i have got this far. I have worked out $$ r_1 \times F_1 $$.
So if i do for the rest. Then how do i add then up as it shows in the question?
Am i even right so far what I've done?
I think your equation for $r_3$ should read $+7k$, not $=7k$.
To perform the addition just do vector addition of each of the cross products.