1. **Top-left: Find parametric equation of line through P(-8,5,3) parallel to xy and xz planes.** - A line parallel to the xy-plane has constant $z$. - A line parallel to the xz-plane has constant $y$. Since the line is parallel to both planes, it must be parallel to the $x$-axis (varying $x$, constant $y$ and $z$). Parametric form: $$\{x = -8 + t, y = 5, z = 3\}$$ 2. **Top-right: Find intersection points of lines with coordinate planes.** (a) Line: $x=1-10t$, $y=-10t$, $z=6t+4$ - $xy$-plane: $z=0 \Rightarrow 6t+4=0 \Rightarrow t=-\frac{2}{3}$ $$x=1-10(-\frac{2}{3})=1+\frac{20}{3}=\frac{23}{3}, y=-10(-\frac{2}{3})=\frac{20}{3}, z=0$$ - $yz$-plane: $x=0 \Rightarrow 1-10t=0 \Rightarrow t=\frac{1}{10}$ $$x=0, y=-10(\frac{1}{10})=-1, z=6(\frac{1}{10})+4=\frac{6}{10}+4=\frac{26}{5}$$ - $xz$-plane: $y=0 \Rightarrow -10t=0 \Rightarrow t=0$ $$x=1, y=0, z=4$$ (b) $\mathbf{r} = 5\mathbf{j} + 4\mathbf{k} + \lambda(3\mathbf{i} - 2\mathbf{j} - 10\mathbf{k})$ - Parametric: $x=3\lambda$, $y=5 - 2\lambda$, $z=4 - 10\lambda$ - $xy$-plane: $z=0 \Rightarrow 4 - 10\lambda=0 \Rightarrow \lambda=\frac{2}{5}$ $$x=3(\frac{2}{5})=\frac{6}{5}, y=5 - 2(\frac{2}{5})=5 - \frac{4}{5}=\frac{21}{5}, z=0$$ - $yz$-plane: $x=0 \Rightarrow 3\lambda=0 \Rightarrow \lambda=0$ $$x=0, y=5, z=4$$ - $xz$-plane: $y=0 \Rightarrow 5 - 2\lambda=0 \Rightarrow \lambda=\frac{5}{2}$ $$x=3(\frac{5}{2})=\frac{15}{2}, y=0, z=4 - 10(\frac{5}{2})=4 - 25 = -21$$ (c) $x=\frac{10}{4}=2.5$, $y + \frac{7}{4} = z = \frac{6}{3} = 2 = \alpha$ - $xy$-plane: $z=0$ but $z=2$ constant, no intersection. - $yz$-plane: $x=0$ but $x=2.5$ constant, no intersection. - $xz$-plane: $y=0$ implies $y + \frac{7}{4} = 0 \Rightarrow y = -\frac{7}{4}$, but $y + \frac{7}{4} = z = 2$, so $z=2$, so point is $(2.5, -\frac{7}{4}, 2)$ 3. **Middle-left: Determine if pairs of lines intersect.** (a) Lines: $x=99-9t$, $y=81-90t$, $z=3t$ $x=27t$, $y=45$, $z=32t$ Set equal: $99-9t=27s$, $81-90t=45$, $3t=32s$ From $y$: $81-90t=45 \Rightarrow 90t=36 \Rightarrow t=\frac{2}{5}$ From $z$: $3(\frac{2}{5})=32s \Rightarrow \frac{6}{5}=32s \Rightarrow s=\frac{3}{80}$ From $x$: $99 - 9(\frac{2}{5})=27(\frac{3}{80}) \Rightarrow 99 - \frac{18}{5} = \frac{81}{80}$ Calculate LHS: $99 - 3.6 = 95.4$; RHS: $1.0125$ no equality, so no intersection. (b) $\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \lambda(7\mathbf{i} - 5\mathbf{j} - 9\mathbf{k})$ $\mathbf{r} = 36\mathbf{i} - 8\mathbf{j} - 333\mathbf{k} + \mu(63\mathbf{i} - 15\mathbf{k})$ Set components equal and solve for $\lambda, \mu$; no consistent solution found, so no intersection. (c) $z + \frac{33}{7} = y + \frac{32}{7} = \frac{10}{3} \alpha$ $z + \frac{565}{70} = \frac{388}{49} = \frac{3.146}{18} \alpha$ No clear parametric form; insufficient info to find intersection. (d) $\mathbf{r} = -54\mathbf{i} + 13\mathbf{j} - 78\mathbf{k} + \lambda(5\mathbf{i} + \mathbf{j} + 7\mathbf{k})$ $\mathbf{r} = -74\mathbf{i} + 24\mathbf{j} + 50\mathbf{k} + \mu(35\mathbf{i} - 3\mathbf{j} - 14\mathbf{k})$ Solve system; no consistent solution, no intersection. (e) Lines: $x=3t-19$, $y=50-8t$, $z=4t-37$ $x=12s+110$, $y=-80s-726$, $z=4s-45$ Set equal: $3t-19=12s+110$, $50-8t=-80s-726$, $4t-37=4s-45$ From $z$: $4t-37=4s-45 \Rightarrow 4t - 4s = -8 \Rightarrow t - s = -2$ From $x$: $3t - 19 = 12s + 110 \Rightarrow 3t - 12s = 129$ Substitute $t = s - 2$: $3(s - 2) - 12s = 129 \Rightarrow 3s - 6 - 12s = 129 \Rightarrow -9s = 135 \Rightarrow s = -15$ Then $t = -15 - 2 = -17$ Check $y$: $50 - 8(-17) = 50 + 136 = 186$ $-80(-15) - 726 = 1200 - 726 = 474$ Not equal, so no intersection. 4. **Middle-right: Parametric equation of line through P1(-5, -1/2, 0) and P2(4, 1, 0).** Direction vector: $$\vec{d} = (4 - (-5), 1 - (-\frac{1}{2}), 0 - 0) = (9, \frac{3}{2}, 0)$$ Parametric form: $$\{x = -5 + 9t, y = -\frac{1}{2} + \frac{3}{2}t, z = 0\}$$ 5. **Bottom-left: Vector equation of line through P1(6, -1, 1) and P2(1, 3, -5).** Direction vector: $$\vec{d} = (1 - 6, 3 - (-1), -5 - 1) = (-5, 4, -6)$$ Vector equation: $$\mathbf{r} = 6\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-5\mathbf{i} + 4\mathbf{j} - 6\mathbf{k})$$ 6. **Bottom-right: Parametric equation of line through P1(\frac{3}{2}, 2, 2) parallel to line given by** $$\frac{x - \frac{1}{2}}{3} = \frac{y - 2}{-2} = \frac{z - 1}{-3} = t$$ Direction vector of given line: $$(3, -2, -3)$$ Parametric form of required line: $$\{x = \frac{3}{2} + 3\alpha, y = 2 - 2\alpha, z = 2 - 3\alpha\}$$