Determine whether the following pairs of planes are coincident, parallel and distinct or neither.

a) x + 3y - z - 2 = 0 and 2x + 6y - 2z - 8 = 0

TEXTBOOK ANSWER: parallel and distinct

Can someone show me how to find the vector equation or parametric equation from the scalar equation? Show me in steps please?

Thanks

To determine whether two planes are coincident, parallel and distinct, or neither, you can examine the coefficients of the variables in the equations of the planes.

Given the equations of the planes:

a) x + 3y - z - 2 = 0 and 2x + 6y - 2z - 8 = 0

To find the vector equation or parametric equation from the scalar equation, we need to express the equations in a form that shows the normal vectors of the planes.

1. Convert the equations to the form Ax + By + Cz + D = 0.

For the first equation, x + 3y - z - 2 = 0, we have the coefficients A = 1, B = 3, C = -1, and D = -2.

For the second equation, 2x + 6y - 2z - 8 = 0, we have the coefficients A = 2, B = 6, C = -2, and D = -8.

2. The normal vector of a plane can be obtained by taking the coefficients of x, y, and z in the equation. So, the normal vector for the first plane is N₁ = [A, B, C] = [1, 3, -1], and for the second plane, N₂ = [2, 6, -2].

3. Now, check if the normal vectors are parallel. If two vectors are parallel, their ratio of corresponding components must be equal. In this case, compare the components of N₁ and N₂:

N₁ / N₂ = [1/2, 3/6, -1/-2] = [1/2, 1/2, 1/2]

Since the ratio of the corresponding components is the same, the normal vectors are parallel.

4. Finally, determine if the planes are coincident, parallel and distinct, or neither:

- If the normal vectors are parallel and the planes do not intersect, they are parallel and distinct.
- If the normal vectors are parallel and the planes coincide with each other, they are coincident.
- If the normal vectors are neither parallel nor coincident, the planes are neither coincident nor parallel.

In this case, the normal vectors are parallel, but the planes have different distances from the origin (different D values). Therefore, the planes are parallel and distinct.

Remember to check the textbook answer to make sure it aligns with the steps you took.