# Vector Calculations Given vectors: \[ \vec{M} = 2\hat{x} - 3\hat{y} + \hat{z} \] \[ \vec{N} = 4\hat{x} + 5\hat{y} - 2\hat{z} \] --- ## a. Find \(\vec{M} \times \vec{N}\) ### Cross product formula: \[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \\ \end{vmatrix} \] ### Components: \[ \vec{M} = (2, -3, ) \] \[ \vec{N} = (4, 5, -2) \] ### Calculation: \[ \vec{M} \times \vec{N} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ 2 & -3 & \\ 4 & 5 & -2 \\ \end{vmatrix} \] \[ = \hat{x}((-3)(-2) - \times 5) - \hat{y}(2 \times (-2) - \times 4) + \hat{z}(2 \times 5 - (-3) \times 4) \] \[ = \hat{x}(6 - ) - \hat{y}(-4 - ) + \hat{z}(10 - (-12)) \] \[ = 6\hat{x} + 4\hat{y} + 22\hat{z} \] **Answer:** \[ \boxed{ \vec{M} \times \vec{N} = 6\hat{x} + 4\hat{y} + 22\hat{z} } \] --- ## b. Show that \(\vec{M} \times \vec{N}\) is perpendicular to both \(\vec{M}\) and \(\vec{N}\) ### Dot products: - \(\vec{M} \cdot (\vec{M} \times \vec{N})\) - \(\vec{N} \cdot (\vec{M} \times \vec{N})\) ### Calculate: \[ \vec{M} = (2, -3, ) \] \[ \vec{M} \times \vec{N} = (6, 4, 22) \] \[ \vec{M} \cdot (\vec{M} \times \vec{N}) = 2 \times 6 + (-3) \times 4 + \times 22 = 12 - 12 + = \] Similarly, for \(\vec{N}\): \[ \vec{N} = (4, 5, -2) \] \[ \vec{N} \cdot (\vec{M} \times \vec{N}) = 4 \times 6 + 5 \times 4 + (-2) \times 22 = 24 + 20 - 44 = \] ### Conclusion: \[ \boxed{ \text{The cross product is perpendicular to both } \vec{M} \text{ and } \vec{N} } \]