What is the linear velocity, if angular velocity vector $$\omega = 3\hat i - 4\hat j + \hat k$$    and position vector $$r = 5\hat i - 6\hat j + 6\hat k?$$

Solution :
The relation between linear velocity $$v,$$ angular velocity $$\omega $$ and position vector $$r$$ is given by
$$\eqalign{ & v = \omega \times r \cr & = \left( {3\hat i - 4\hat j + \hat k} \right) \times \left( {5\hat i - 6\hat j + 6\hat k} \right) \cr} $$
\[\begin{array}{l} = \left| {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k}\\ 3&{ - 4}&1\\ 5&{ - 6}&6 \end{array}} \right|\\ = \hat i\left| {\begin{array}{*{20}{c}} { - 4}&1\\ { - 6}&6 \end{array}} \right| - \hat j\left| {\begin{array}{*{20}{c}} 3&1\\ 5&6 \end{array}} \right| + \hat k\left| {\begin{array}{*{20}{c}} 3&{ - 4}\\ 5&{ - 6} \end{array}} \right| \end{array}\]
$$\eqalign{ & = \left( { - 24 + 6} \right)\hat i - \left( {18 - 5} \right)\hat j + \left( { - 18 + 20} \right)\hat k \cr & = - 18\hat i - 13\hat j + 2\hat k \cr} $$
Alternative
$$\eqalign{ & v = \omega \times r \cr & = \left( {3\hat i - 4\hat j + \hat k} \right) \times \left( {5\hat i - 6\hat j + 6\hat k} \right) \cr & = \left( {3 \times 5} \right)\left( {\hat i \times \hat i} \right) + \left[ {3 \times \left( { - 6} \right)} \right]\left( {\hat i \times \hat j} \right) + \left( {3 \times 6} \right)\left( {\hat i \times \hat k} \right) + \left( { - 4 \times 5} \right)\left( {\hat j \times \hat i} \right) + \left( { - 4 \times - 6} \right)\left( {\hat j \times \hat j} \right) + \left( { - 4 \times 6} \right)\left( {\hat j \times \hat k} \right) + \left( {1 \times 5} \right)\left( {\hat k \times \hat i} \right) + \left( {1 \times - 6} \right)\left( {\hat k \times \hat j} \right) + \left( {1 \times 6} \right)\left( {\hat k \times \hat k} \right) \cr & {\text{Use}}\,\,\hat i \times \hat j = - \hat j \times \hat i = \hat k \cr & \hat j \times \hat k = - \hat k \times \hat j = \hat i \cr & {\text{and}}\,\,\hat k \times \hat i = - \hat i \times \hat k = \hat j \cr & {\text{Thus,}}\,v = 0 + \left( { - 18} \right)\left( {\hat k} \right) + \left( {18} \right)\left( { - \hat j} \right) + \left( { - 20} \right)\left( { - \hat k} \right) + 0 + \left( { - 24} \right)\left( {\hat i} \right) + \left( 5 \right)\left( {\hat j} \right) + \left( { - 6} \right)\left( { - \hat i} \right) + 0 \cr & = - 18\hat k - 18\hat j + 20\hat k - 24\hat i + 5\hat j + 6\hat i \cr & = - 18\hat i - 13\hat j + 2\hat k \cr} $$