To show that two planes are parallel in vector form, you need to determine whether their normal vectors are scalar multiples of each other. Here's the step-by-step process:

1. Write the equations of the planes in vector form:

A plane is typically expressed as:

$a_1x + b_1y + c_1z + d_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2z + d_2 = 0$

Here:

• $\mathbf{n}_1 = \langle a_1, b_1, c_1 \rangle$ is the normal vector for the first plane.
• $\mathbf{n}_2 = \langle a_2, b_2, c_2 \rangle$ is the normal vector for the second plane.

2. Compare the normal vectors:

If the two planes are parallel, their normal vectors must be scalar multiples of each other:

$\mathbf{n}_1 = k \mathbf{n}_2 \quad \text{for some scalar } k$

This means: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$

If these ratios are equal, then the planes are parallel.

3. Verify consistency of the planes (optional):

Even if the normal vectors are scalar multiples, the planes may be distinct (not coincident). You can determine if they are distinct or coincident by checking if substituting a point from one plane into the other satisfies the equation.

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Example:

Let’s consider two planes:

1. $2x - y + z + 3 = 0$
2. $4x - 2y + 2z - 6 = 0$

• The normal vectors are $\mathbf{n}_1 = \langle 2, -1, 1 \rangle$ and $\mathbf{n}_2 = \langle 4, -2, 2 \rangle$.
• Check if $\mathbf{n}_1$ and $\mathbf{n}_2$ are scalar multiples:

$\frac{4}{2} = \frac{-2}{-1} = \frac{2}{1} = 2$

Thus, $\mathbf{n}_1 = 2 \mathbf{n}_2$, so the planes are parallel.

If you'd like further clarification or examples, let me know! Here's how you can explore this concept:

Questions for Exploration:

1. How do you determine if planes are coincident after proving they are parallel?
2. What happens if one normal vector is the zero vector?
3. Can two planes be neither parallel nor perpendicular? How would that look?
4. How do parallel planes appear geometrically in 3D space?
5. How can the distance between two parallel planes be calculated?

Tip:

Always check for consistency (substitute a point from one plane into the