1. **Problem statement:** We have four vectors normal to surfaces at point P and four tangent plane equations at P. We need to match each vector and each plane equation to one of four diagrams (A, B, C, D) based on the orientation of level curves and the point P. 2. **Step 1: Understand normal vectors and tangent planes** - The normal vector to the surface at P is perpendicular to the tangent plane at P. - The tangent plane equation can be written as $a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$ where $(a,b,c)$ is the normal vector. 3. **Step 2: Analyze given normal vectors** - Vector 1: $2\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}$ corresponds to normal $(2,2,-2)$ - Vector 2: $2\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ corresponds to normal $(2,-2,2)$ - Vector 3: $2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ corresponds to normal $(2,2,2)$ - Vector 4: $-2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ corresponds to normal $(-2,2,2)$ 4. **Step 3: Analyze tangent plane equations and extract normals** - Plane 1: $2x - 2y - 2z = 2$ normal vector $(2,-2,-2)$ - Plane 2: $-\frac{x}{2} + \frac{y}{2} - \frac{z}{2} = -7$ normal vector $(-\frac{1}{2}, \frac{1}{2}, -\frac{1}{2})$ which is proportional to $(-1,1,-1)$ - Plane 3: $-3x - 3y + 3z = 6$ normal vector $(-3,-3,3)$ proportional to $(-1,-1,1)$ - Plane 4: $x + y + z = 4$ normal vector $(1,1,1)$ 5. **Step 4: Match vectors to planes by direction** - Vector 1 $(2,2,-2)$ matches Plane 1 $(2,-2,-2)$? No, signs differ. - Vector 2 $(2,-2,2)$ matches Plane 2 $(-1,1,-1)$? No, signs differ. - Vector 3 $(2,2,2)$ matches Plane 4 $(1,1,1)$ yes, same direction. - Vector 4 $(-2,2,2)$ matches Plane 3 $(-1,-1,1)$? No, signs differ. 6. **Step 5: Consider sign flips (normal vectors can be reversed)** - Vector 1 $(2,2,-2)$ and Plane 2 $(-1,1,-1)$: no match. - Vector 1 and Plane 1 $(2,-2,-2)$: signs differ in y. - Vector 2 $(2,-2,2)$ and Plane 1 $(2,-2,-2)$: differ in z. - Vector 2 and Plane 2 $(-1,1,-1)$: differ in all signs. - Vector 4 $(-2,2,2)$ and Plane 3 $(-1,-1,1)$: differ in y. 7. **Step 6: Use level curve orientation to match diagrams** - Diagram A: level curves low to high along concentric arcs, P center right. - Diagram B: level curves high to low bottom left to top right, P center right. - Diagram C: level curves high to low top left to bottom right, P top left. - Diagram D: level curves low to high top left to bottom right, P top left. 8. **Step 7: Assign matches based on normal vector directions and level curve gradients** - Vector 3 $(2,2,2)$ and Plane 4 $(1,1,1)$ match Diagram A (concentric increasing curves, normal points outward). - Vector 1 $(2,2,-2)$ and Plane 1 $(2,-2,-2)$ match Diagram D (low to high top left to bottom right). - Vector 2 $(2,-2,2)$ and Plane 2 $(-1,1,-1)$ match Diagram B (high to low bottom left to top right). - Vector 4 $(-2,2,2)$ and Plane 3 $(-1,-1,1)$ match Diagram C (high to low top left to bottom right). **Final matches:** **(a) Normal vectors to diagrams:** - 1: Diagram D - 2: Diagram B - 3: Diagram A - 4: Diagram C **(b) Tangent planes to diagrams:** - 1: Diagram D - 2: Diagram B - 3: Diagram C - 4: Diagram A