1. **State the problem:** Find the angle between two lines given in various forms (parametric, vector, symmetric). 2. **Formula:** The angle $\theta$ between two lines with direction vectors $\mathbf{a}$ and $\mathbf{b}$ is given by: $$\cos \theta = \frac{|\mathbf{a} \cdot \mathbf{b}|}{\|\mathbf{a}\| \|\mathbf{b}\|}$$ where $\mathbf{a} \cdot \mathbf{b}$ is the dot product and $\|\mathbf{a}\|$ is the magnitude of vector $\mathbf{a}$. 3. **Extract direction vectors:** - For parametric lines, coefficients of $t$ or $\lambda$ give direction vectors. - For symmetric form, denominators give direction vectors. 4. **Calculate angles for each pair:** **Pair 1:** $x=9-6t, y=-5t-10, z=0$ direction vector $\mathbf{a} = (-6, -5, 0)$ $x=-5t-7, y=2t, z=-4t-5$ direction vector $\mathbf{b} = (-5, 2, -4)$ Dot product: $$\mathbf{a} \cdot \mathbf{b} = (-6)(-5) + (-5)(2) + 0(-4) = 30 - 10 + 0 = 20$$ Magnitudes: $$\|\mathbf{a}\| = \sqrt{(-6)^2 + (-5)^2 + 0^2} = \sqrt{36 + 25} = \sqrt{61}$$ $$\|\mathbf{b}\| = \sqrt{(-5)^2 + 2^2 + (-4)^2} = \sqrt{25 + 4 + 16} = \sqrt{45}$$ Cosine: $$\cos \theta = \frac{20}{\sqrt{61} \times \sqrt{45}} = \frac{20}{\sqrt{2745}}$$ Angle: $$\theta = \cos^{-1}\left(\frac{20}{\sqrt{2745}}\right) \approx 68^\circ$$ **Pair 2:** $r = \mathbf{i} - 4\mathbf{k} + \lambda (3\mathbf{k})$ direction vector $\mathbf{a} = (0, 0, 3)$ $r = 3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k} + \lambda (-2\mathbf{i} + 8\mathbf{j} + 9\mathbf{k})$ direction vector $\mathbf{b} = (-2, 8, 9)$ Dot product: $$\mathbf{a} \cdot \mathbf{b} = 0(-2) + 0(8) + 3(9) = 27$$ Magnitudes: $$\|\mathbf{a}\| = 3$$ $$\|\mathbf{b}\| = \sqrt{(-2)^2 + 8^2 + 9^2} = \sqrt{4 + 64 + 81} = \sqrt{149}$$ Cosine: $$\cos \theta = \frac{27}{3 \times \sqrt{149}} = \frac{9}{\sqrt{149}}$$ Angle: $$\theta = \cos^{-1}\left(\frac{9}{\sqrt{149}}\right) \approx 44^\circ$$ **Pair 3:** Symmetric form: $\frac{x-8}{-3} = \frac{y+8}{-8} = \frac{z+5}{-10} = \alpha$ direction vector $\mathbf{a} = (-3, -8, -10)$ $\frac{x+7}{9} = \frac{y-5}{-8} = \frac{z+6}{-7} = \alpha$ direction vector $\mathbf{b} = (9, -8, -7)$ Dot product: $$\mathbf{a} \cdot \mathbf{b} = (-3)(9) + (-8)(-8) + (-10)(-7) = -27 + 64 + 70 = 107$$ Magnitudes: $$\|\mathbf{a}\| = \sqrt{9 + 64 + 100} = \sqrt{173}$$ $$\|\mathbf{b}\| = \sqrt{81 + 64 + 49} = \sqrt{194}$$ Cosine: $$\cos \theta = \frac{107}{\sqrt{173} \times \sqrt{194}}$$ Angle: $$\theta = \cos^{-1}\left(\frac{107}{\sqrt{173 \times 194}}\right) \approx 18^\circ$$ **Pair 4:** $r = 10\mathbf{i} - 5\mathbf{j} - 4\mathbf{k} + \lambda (-4\mathbf{i} + 2\mathbf{j} - 8\mathbf{k})$ direction vector $\mathbf{a} = (-4, 2, -8)$ $r = 10\mathbf{i} - 4\mathbf{j} - 8\mathbf{k} + \lambda (-4\mathbf{i} - 3\mathbf{j})$ direction vector $\mathbf{b} = (-4, -3, 0)$ Dot product: $$\mathbf{a} \cdot \mathbf{b} = (-4)(-4) + 2(-3) + (-8)(0) = 16 - 6 + 0 = 10$$ Magnitudes: $$\|\mathbf{a}\| = \sqrt{16 + 4 + 64} = \sqrt{84}$$ $$\|\mathbf{b}\| = \sqrt{16 + 9 + 0} = \sqrt{25} = 5$$ Cosine: $$\cos \theta = \frac{10}{\sqrt{84} \times 5} = \frac{10}{5\sqrt{84}} = \frac{2}{\sqrt{84}}$$ Angle: $$\theta = \cos^{-1}\left(\frac{2}{\sqrt{84}}\right) \approx 76^\circ$$ **Pair 5:** $x=9t-6, y=4t+10, z=-4$ direction vector $\mathbf{a} = (9, 4, 0)$ $x=9t+10, y=7t-7, z=-7$ direction vector $\mathbf{b} = (9, 7, 0)$ Dot product: $$\mathbf{a} \cdot \mathbf{b} = 9(9) + 4(7) + 0(0) = 81 + 28 = 109$$ Magnitudes: $$\|\mathbf{a}\| = \sqrt{81 + 16 + 0} = \sqrt{97}$$ $$\|\mathbf{b}\| = \sqrt{81 + 49 + 0} = \sqrt{130}$$ Cosine: $$\cos \theta = \frac{109}{\sqrt{97} \times \sqrt{130}}$$ Angle: $$\theta = \cos^{-1}\left(\frac{109}{\sqrt{97 \times 130}}\right) \approx 14^\circ$$ **Final answers:** - Pair 1: $68^\circ$ - Pair 2: $44^\circ$ - Pair 3: $18^\circ$ - Pair 4: $76^\circ$ - Pair 5: $14^\circ$