Consider the plane, \( P \), which passes through points \( (0,0,0),(1,2,3) \) and \( (3,2,1) \). Which of the following planes is parallel to \( P \) ? The plane with equation \( -4 x+8 y-4 z=10 \) The plane with equation \( x+2 y+3 z=0 \) The xy-plane The plane with equation \( 4 x+8 y+4 z=10 \)

To determine which of the given planes is parallel to the plane \( P \) passing through points \( (0,0,0), (1,2,3) \), and \( (3,2,1) \), we need to find the normal vector of plane \( P \) and then compare it with the normal vectors of the given planes. 1. Find the normal vector of plane \( P \): The normal vector of a plane passing through three points \( (x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \) is given by the cross product of the vectors formed by these points: \[ \vec{n} = \vec{a} \times \vec{b} \] where \( \vec{a} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \) and \( \vec{b} = (x_3 - x_1, y_3 - y_1, z_3 - z_1) \). Let's calculate the normal vector of plane \( P \) using the given points \( (0,0,0), (1,2,3) \), and \( (3,2,1) \). Calculate the value by following steps: - step0: Calculate: \(\left(1-0.2-0.3-0\right)\left(3-0.2-0.1-0\right)\) - step1: Remove 0: \(\left(1-0.2-0.3-0\right)\left(3-0.2-0.1\right)\) - step2: Remove 0: \(\left(1-0.2-0.3\right)\left(3-0.2-0.1\right)\) - step3: Subtract the numbers: \(0.5\left(3-0.2-0.1\right)\) - step4: Subtract the numbers: \(0.5\times 2.7\) - step5: Multiply the numbers: \(1.35\) The normal vector of plane \( P \) is \( \left(\frac{27}{20}, \frac{7}{20}, 1\right) \). 2. Compare the normal vectors of the given planes with the normal vector of plane \( P \): - The plane with equation \( -4x + 8y - 4z = 10 \) has a normal vector of \( \langle -4, 8, -4 \rangle \). - The plane with equation \( x + 2y + 3z = 0 \) has a normal vector of \( \langle 1, 2, 3 \rangle \). - The xy-plane has a normal vector of \( \langle 0, 0, 1 \rangle \). - The plane with equation \( 4x + 8y + 4z = 10 \) has a normal vector of \( \langle 4, 8, 4 \rangle \). Comparing the normal vectors, we can see that the plane with equation \( 4x + 8y + 4z = 10 \) has a normal vector that is parallel to the normal vector of plane \( P \). Therefore, the plane with equation \( 4x + 8y + 4z = 10 \) is parallel to plane \( P \).