Question

Evaluate: \[ (\vec{i} - 3\vec{j} + 4\vec{k}) \cdot \big[(2\vec{i} - \vec{j}) \times (\vec{j} + \vec{k})\big]. \]

Hint:

Calculate cross products using the determinant of a 3x3 matrix; use dot product formula for final evaluation.

Answer 1

First, find the cross product: \[ (2\vec{i} - \vec{j}) \times (\vec{j} + \vec{k}). \] Calculate: \[ = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\ 2 & -1 & 0 \\ 0 & 1 & 1 \end{vmatrix} = \vec{i} \begin{vmatrix} -1 & 0 \\ 1 & 1 \end{vmatrix} - \vec{j} \begin{vmatrix} 2 & 0 \\ 0 & 1 \end{vmatrix} + \vec{k} \begin{vmatrix} 2 & -1 \\ 0 & 1 \end{vmatrix}. \] Evaluate minors: \[ = \vec{i}((-1)(1) - (0)(1)) - \vec{j}(2 \cdot 1 - 0 \cdot 0) + \vec{k}(2 \cdot 1 - 0 \cdot (-1)) \] \[ = \vec{i}(-1) - \vec{j}(2) + \vec{k}(2) = -\vec{i} - 2\vec{j} + 2\vec{k}. \] Now, take the dot product: \[ (\vec{i} - 3\vec{j} + 4\vec{k}) \cdot (-\vec{i} - 2\vec{j} + 2\vec{k}) = (1)(-1) + (-3)(-2) + (4)(2). \] Calculate: \[ = -1 + 6 + 8 = 13. \] \[ \boxed{13}. \]