Questions: Find the length and direction (when defined) of u × v and v × u. u = -4i - 4j - 6k, v = 5i + 5j + 3k

Solution Steps

Given vectors: \[ \mathbf{u} = -4 \mathbf{i} - 4 \mathbf{j} - 6 \mathbf{k} \] \[ \mathbf{v} = 5 \mathbf{i} + 5 \mathbf{j} + 3 \mathbf{k} \]

The cross product \(\mathbf{u} \times \mathbf{v}\) is: \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -4 & -4 & -6 \\ 5 & 5 & 3 \end{vmatrix} = 18 \mathbf{i} - 18 \mathbf{j} + 0 \mathbf{k} = [18, -18, 0] \]

The cross product \(\mathbf{v} \times \mathbf{u}\) is: \[ \mathbf{v} \times \mathbf{u} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 5 & 3 \\ -4 & -4 & -6 \end{vmatrix} = -18 \mathbf{i} + 18 \mathbf{j} + 0 \mathbf{k} = [-18, 18, 0] \]

The length of \(\mathbf{u} \times \mathbf{v}\) is: \[ \|\mathbf{u} \times \mathbf{v}\| = \sqrt{18^2 + (-18)^2 + 0^2} = \sqrt{324 + 324} = \sqrt{648} \approx 25.4558 \]

The length of \(\mathbf{v} \times \mathbf{u}\) is: \[ \|\mathbf{v} \times \mathbf{u}\| = \sqrt{(-18)^2 + 18^2 + 0^2} = \sqrt{324 + 324} = \sqrt{648} \approx 25.4558 \]

The direction of \(\mathbf{u} \times \mathbf{v}\) is: \[ \frac{\mathbf{u} \times \mathbf{v}}{\|\mathbf{u} \times \mathbf{v}\|} = \frac{[18, -18, 0]}{25.4558} \approx [0.7071, -0.7071, 0] \]

The direction of \(\mathbf{v} \times \mathbf{u}\) is: \[ \frac{\mathbf{v} \times \mathbf{u}}{\|\mathbf{v} \times \mathbf{u}\|} = \frac{[-18, 18, 0]}{25.4558} \approx [-0.7071, 0.7071, 0] \]

Final Answer