1. **Problem 25:** Find a unit vector in the same direction as the vector $\langle 8, -1, 4 \rangle$. 2. **Formula:** A unit vector $\mathbf{u}$ in the direction of vector $\mathbf{v}$ is given by $$\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$$ where $\|\mathbf{v}\|$ is the magnitude (length) of $\mathbf{v}$. 3. **Calculate the magnitude of $\mathbf{v} = \langle 8, -1, 4 \rangle$**: $$\|\mathbf{v}\| = \sqrt{8^2 + (-1)^2 + 4^2} = \sqrt{64 + 1 + 16} = \sqrt{81} = 9$$ 4. **Find the unit vector:** $$\mathbf{u} = \frac{1}{9} \langle 8, -1, 4 \rangle = \left\langle \frac{8}{9}, \frac{-1}{9}, \frac{4}{9} \right\rangle$$ --- 5. **Problem 26:** Find the vector in the same direction as $\langle 6, 2, -3 \rangle$ but with length 4. 6. **Formula:** To find a vector $\mathbf{w}$ with length $L$ in the direction of $\mathbf{v}$, $$\mathbf{w} = L \cdot \frac{\mathbf{v}}{\|\mathbf{v}\|}$$ 7. **Calculate the magnitude of $\mathbf{v} = \langle 6, 2, -3 \rangle$**: $$\|\mathbf{v}\| = \sqrt{6^2 + 2^2 + (-3)^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7$$ 8. **Find the vector with length 4:** $$\mathbf{w} = 4 \cdot \frac{1}{7} \langle 6, 2, -3 \rangle = \left\langle \frac{24}{7}, \frac{8}{7}, \frac{-12}{7} \right\rangle$$