1. **State the problem:** Determine which pairs of the given planes are orthogonal (perpendicular) and which are parallel. 2. **Recall the formula and rules:** - The normal vector of a plane $Ax + By + Cz + D = 0$ is $\vec{n} = (A, B, C)$. - Two planes are **parallel** if their normal vectors are scalar multiples: $\vec{n_1} = k \vec{n_2}$ for some scalar $k$. - Two planes are **orthogonal** if their normal vectors are perpendicular, i.e., their dot product is zero: $\vec{n_1} \cdot \vec{n_2} = 0$. 3. **Extract normal vectors:** - Plane 1: $\vec{n_1} = (-9, 2, -7)$ - Plane 2: $\vec{n_2} = (\frac{5}{4}, -\frac{5}{4}, -\frac{5}{4})$ - Plane 3: $\vec{n_3} = (-1, 1, 1)$ - Plane 4: $\vec{n_4} = (4, 5, \frac{49}{5})$ - Plane 5: $\vec{n_5} = (-2, -4, \frac{10}{7})$ - Plane 6: $\vec{n_6} = (2, -2, -\frac{6}{5})$ - Plane 7: $\vec{n_7} = (10, 4, 10)$ - Plane 8: $\vec{n_8} = (-16, 24, -8)$ - Plane 9: $\vec{n_9} = (2, -3, 1)$ - Plane 10: $\vec{n_{10}} = (-6, -5, 5)$ 4. **Check parallelism:** - Compare vectors to see if one is scalar multiple of another. - Pairs found parallel: - Planes 3 and 4: $\vec{n_4} = (4,5,9.8)$ and $\vec{n_3} = (-1,1,1)$ are not scalar multiples. - Planes 7 and 8: $\vec{n_7} = (10,4,10)$ and $\vec{n_8} = (-16,24,-8)$ no scalar multiple. - Planes 5 and 6: $\vec{n_5} = (-2,-4,1.4286)$ and $\vec{n_6} = (2,-2,-1.2)$ no scalar multiple. - Planes 9 and 10: $\vec{n_9} = (2,-3,1)$ and $\vec{n_{10}} = (-6,-5,5)$ no scalar multiple. - Planes 1 and 2: $\vec{n_1} = (-9,2,-7)$ and $\vec{n_2} = (1.25,-1.25,-1.25)$ no scalar multiple. 5. **Check orthogonality:** - Compute dot products: - $\vec{n_1} \cdot \vec{n_2} = (-9)(1.25) + 2(-1.25) + (-7)(-1.25) = -11.25 - 2.5 + 8.75 = -5$ - $\vec{n_3} \cdot \vec{n_4} = (-1)(4) + 1(5) + 1(9.8) = -4 + 5 + 9.8 = 10.8$ - $\vec{n_5} \cdot \vec{n_6} = (-2)(2) + (-4)(-2) + 1.4286(-1.2) = -4 + 8 - 1.7143 = 2.2857$ - $\vec{n_7} \cdot \vec{n_8} = 10(-16) + 4(24) + 10(-8) = -160 + 96 - 80 = -144$ - $\vec{n_9} \cdot \vec{n_{10}} = 2(-6) + (-3)(-5) + 1(5) = -12 + 15 + 5 = 8$ - None of these dot products are zero, so no orthogonal pairs found. 6. **Re-examine for errors:** - Normalize vectors and check again for parallelism and orthogonality. - Actually, planes 1 and 2: - $\vec{n_1} = (-9, 2, -7)$ - $\vec{n_2} = (1.25, -1.25, -1.25)$ - Dot product: $-9(1.25) + 2(-1.25) + (-7)(-1.25) = -11.25 - 2.5 + 8.75 = -5$ - Not zero, so not orthogonal. - Check if any pairs are scalar multiples: - Planes 7 and 8: - $\vec{n_7} = (10,4,10)$ - $\vec{n_8} = (-16,24,-8)$ - Ratios: $-16/10 = -1.6$, $24/4=6$, $-8/10=-0.8$ not equal, so not parallel. 7. **Final conclusion:** - No pairs are parallel or orthogonal based on normal vectors. **Answer:** - **Perpendicular planes:** $\emptyset$ - **Parallel planes:** $\emptyset$