Example

Question:

Find the scalar and vector products of two vectors.
\( \mathbf{a} = 3\mathbf{\hat{i}} - 4\mathbf{\hat{j}} + 5\mathbf{\hat{k}} \)\,
\( \mathbf{b} = -2\mathbf{\hat{i}} + \mathbf{\hat{j}} - 3\mathbf{\hat{k}} \)

Solution:

Scalar (dot) product:
\[ \mathbf{a} \cdot \mathbf{b} = (3\mathbf{\hat{i}} - 4\mathbf{\hat{j}} + 5\mathbf{\hat{k}}) \cdot (-2\mathbf{\hat{i}} + \mathbf{\hat{j}} - 3\mathbf{\hat{k}}) \] Calculate each term:
\[ = (3 \times -2) + (-4 \times 1) + (5 \times -3) = -6 - 4 - 15 = -25 \]
Vector (cross) product:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ 3 & -4 & 5 \\ -2 & 1 & -3 \\ \end{vmatrix} = 7\mathbf{\hat{i}} - \mathbf{\hat{j}} - 5\mathbf{\hat{k}} \]
Note: \(\mathbf{b} \times \mathbf{a} = -7\mathbf{\hat{i}} + \mathbf{\hat{j}} + 5\mathbf{\hat{k}} \)