What is the cross product a x b for the vectors a = (-2i + 3j + 4k) and b = (3i + 2k)?

To calculate the cross product of the vectors a and b, we can use the determinant method with the following formulation:

Given vectors: a = -2i + 3j + 4k b = 3i + 0j + 2k

We can express this in a determinant form: [ a \times b = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ -2 & 3 & 4 \ 3 & 0 & 2 \end{vmatrix} ]

Expanding this determinant, we calculate it step by step:

1. For the i-component: [ \hat{i} \left( (3)(2) - (4)(0) \right) = \hat{i} (6 - 0) = 6\hat{i} ]

2. For the j-component: [ -\hat{j} \left( (-2)(2) - (4)(3) \right) = -\hat{j} (-4 - 12) = -\hat{j} (-16) = 16\hat{j} ]

3. For the k