Standard +0.8 This M5 question requires verifying forces sum to zero (straightforward vector addition), then computing three 3D cross products r×F and summing them to find the couple's moment magnitude. While mechanically intensive with multiple cross products and careful arithmetic, it's a standard application of rigid body mechanics without novel insight—typical for Further Maths M5 but more demanding than average A-level questions due to computational complexity and 3D vector operations.
Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) act on a rigid body at the points with position vectors \(\mathbf{r}_1\), \(\mathbf{r}_2\) and \(\mathbf{r}_3\) respectively.
\(\mathbf{F}_1 = (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\) N and \(\mathbf{r}_1 = (\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) m,
\(\mathbf{F}_2 = (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k})\) N and \(\mathbf{r}_2 = (3\mathbf{i} - \mathbf{j} - \mathbf{k})\) m,
\(\mathbf{F}_3 = (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) N and \(\mathbf{r}_3 = (\mathbf{i} - 2\mathbf{j} + \mathbf{k})\) m.
Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple.
[9]
Three forces $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ act on a rigid body at the points with position vectors $\mathbf{r}_1$, $\mathbf{r}_2$ and $\mathbf{r}_3$ respectively.
$\mathbf{F}_1 = (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})$ N and $\mathbf{r}_1 = (\mathbf{i} + \mathbf{j} - 2\mathbf{k})$ m,
$\mathbf{F}_2 = (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k})$ N and $\mathbf{r}_2 = (3\mathbf{i} - \mathbf{j} - \mathbf{k})$ m,
$\mathbf{F}_3 = (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k})$ N and $\mathbf{r}_3 = (\mathbf{i} - 2\mathbf{j} + \mathbf{k})$ m.
Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple.
[9]
\hfill \mbox{\textit{Edexcel M5 2014 Q3 [9]}}