1. **State the problem:** Express the vector $\mathbf{v} = 5\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$ as a sum of two vectors, one parallel and one perpendicular to $\mathbf{u} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$. 2. **Formula and rules:** - The vector parallel to $\mathbf{u}$ is the projection of $\mathbf{v}$ onto $\mathbf{u}$: $$\mathbf{v}_{\parallel} = \mathrm{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$$ - The vector perpendicular to $\mathbf{u}$ is: $$\mathbf{v}_{\perp} = \mathbf{v} - \mathbf{v}_{\parallel}$$ 3. **Calculate dot products:** $$\mathbf{v} \cdot \mathbf{u} = (5)(2) + (2)(-1) + (-3)(3) = 10 - 2 - 9 = -1$$ $$\mathbf{u} \cdot \mathbf{u} = (2)^2 + (-1)^2 + (3)^2 = 4 + 1 + 9 = 14$$ 4. **Find the parallel vector:** $$\mathbf{v}_{\parallel} = \frac{-1}{14} (2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = \left(-\frac{2}{14}\right)\mathbf{i} + \frac{1}{14}\mathbf{j} - \frac{3}{14}\mathbf{k} = -\frac{1}{7}\mathbf{i} + \frac{1}{14}\mathbf{j} - \frac{3}{14}\mathbf{k}$$ 5. **Find the perpendicular vector:** $$\mathbf{v}_{\perp} = \mathbf{v} - \mathbf{v}_{\parallel} = \left(5 + \frac{1}{7}\right)\mathbf{i} + \left(2 - \frac{1}{14}\right)\mathbf{j} + \left(-3 + \frac{3}{14}\right)\mathbf{k} = \frac{36}{7}\mathbf{i} + \frac{27}{14}\mathbf{j} - \frac{39}{14}\mathbf{k}$$ 6. **Final answer:** $$\mathbf{v} = \mathbf{v}_{\parallel} + \mathbf{v}_{\perp} = \left(-\frac{1}{7}\mathbf{i} + \frac{1}{14}\mathbf{j} - \frac{3}{14}\mathbf{k}\right) + \left(\frac{36}{7}\mathbf{i} + \frac{27}{14}\mathbf{j} - \frac{39}{14}\mathbf{k}\right)$$