Let's solve the two parts of the problem step by step:

Part (a): Minimum Distance Between Two Planes

Given planes:

• $P_1: 2x + 3y - 6z = -10$
• $P_2: 2x + 3y - 6z = 88$

Both planes have the same normal vector $\mathbf{n} = \langle 2, 3, -6 \rangle$. Since the planes are parallel (same normal vector), the minimum distance between them can be calculated using the formula:

$\text{Distance} = \frac{|d_2 - d_1|}{\|\mathbf{n}\|}$

where:

• $d_1 = -10$ (from the equation of $P_1$)
• $d_2 = 88$ (from the equation of $P_2$)
• $\|\mathbf{n}\|$ is the magnitude of the normal vector $\mathbf{n}$.

First, calculate the magnitude of $\mathbf{n}$:

$\|\mathbf{n}\| = \sqrt{2^2 + 3^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$

Now, calculate the distance:

$\text{Distance} = \frac{|88 - (-10)|}{7} = \frac{98}{7} = 14$

So, the minimum distance between the two planes is 14 units.

Part (b): Nearest Point on $P_2$ from Point $(-2, 0, 1)$ on $P_1$

To find the nearest point on $P_2$, we project the point $(-2, 0, 1)$ onto the plane $P_2$.

1. Find the normal vector: $\mathbf{n} = \langle 2, 3, -6 \rangle$.

2. Equation of the line passing through $(-2, 0, 1)$ and along the direction of the normal vector is:

$\mathbf{r}(t) = (-2, 0, 1) + t \langle 2, 3, -6 \rangle = (-2 + 2t, 3t, 1 - 6t)$

3. Substitute these coordinates into the equation of plane $P_2$:

$2(-2 + 2t) + 3(3t) - 6(1 - 6t) = 88$

4. Simplify:

$-4 + 4t + 9t - 6 + 36t = 88$

$49t - 10 = 88$

$49t = 98 \quad \Rightarrow \quad t = 2$

5. Substitute $t = 2$ back into the equation of the line to find the coordinates of the nearest point on $P_2$:

$x = -2 + 2(2) = 2$ $y = 0 + 3(2) = 6$ $z = 1 - 6(2) = -11$

So, the nearest point on $P_2$ is $(2, 6, -11)$.

Summary:

• The minimum distance between the two planes is 14 units.
• The nearest point on $P_2$ to the point $(-2, 0, 1)$ on $P_1$ is $(2, 6, -11)$.

Do you want further details or have any questions?

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Related Questions:

1. How do you determine if two planes are parallel?
2. What is the formula for the distance between a point and a plane?
3. How can you find the distance between two skew lines?
4. What is the geometrical significance of the normal vector of a plane?
5. How do you find the angle between two intersecting planes?

Tip: Always ensure to calculate the magnitude of the normal vector carefully, as it is crucial for determining distances between planes.