Find a point-normal form of the equation of the plane passing through P and having n as a normal.
(a) P(-1,3, -2):n=(-2,1,-1)
(b) P(1, 1,4): n= (1,9, 8)
(c) P(2,0,0):n= (0,0,2)
(d) P(0,0,0):n=(1,2,3)

The **point-normal form** of a plane is given by the equation:

\[
n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0
\]

Where:
- \((x_0, y_0, z_0)\) is the point \(P\) that the plane passes through.
- \(n_1, n_2, n_3\) are the components of the normal vector \(n = (n_1, n_2, n_3)\).

Let's compute the point-normal form for each part:

---

### (a) \( P(-1, 3, -2) \), \( n = (-2, 1, -1) \)

Using the formula:

\[
-2(x + 1) + 1(y - 3) - 1(z + 2) = 0
\]

Simplifying:

\[
-2(x + 1) + (y - 3) - (z + 2) = 0
\]
\[
-2x - 2 + y - 3 - z - 2 = 0
\]
\[
-2x + y - z - 7 = 0
\]

Thus, the point-normal form of the plane is:
\[
-2x + y - z = 7
\]

---

### (b) \( P(1, 1, 4) \), \( n = (1, 9, 8) \)

Using the formula:

\[
1(x - 1) + 9(y - 1) + 8(z - 4) = 0
\]

Simplifying:

\[
(x - 1) + 9(y - 1) + 8(z - 4) = 0
\]
\[
x - 1 + 9y - 9 + 8z - 32 = 0
\]
\[
x + 9y + 8z - 42 = 0
\]

Thus, the point-normal form of the plane is:
\[
x + 9y + 8z = 42
\]

---

### (c) \( P(2, 0, 0) \), \( n = (0, 0, 2) \)

Using the formula:

\[
0(x - 2) + 0(y - 0) + 2(z - 0) = 0
\]

Simplifying:

\[
2(z - 0) = 0
\]
\[
2z = 0
\]

Thus, the point-normal form of the plane is:
\[
z = 0
\]

---

### (d) \( P(0, 0, 0) \), \( n = (1, 2, 3) \)

Using the formula:

\[
1(x - 0) + 2(y - 0) + 3(z - 0) = 0
\]

Simplifying:

\[
x + 2y + 3z = 0
\]

Thus, the point-normal form of the plane is:
\[
x + 2y + 3z = 0
\]

---

### Summary:
- (a) \(-2x + y - z = 7\)
- (b) \(x + 9y + 8z = 42\)
- (c) \(z = 0\)
- (d) \(x + 2y + 3z = 0\)

Do you have any questions or want further details?

Here are some related questions to consider:
1. How do you find the general form of a plane from point-normal form?
2. What happens if the normal vector has all zero components?
3. How do you find the distance from a point to a plane?
4. Can you derive the point of intersection of a line with a plane?
5. What is the relationship between parallel planes?

**Tip:** When converting a point-normal form to other forms of the plane equation, be careful with signs when expanding expressions.

Find a point-normal form of the equation of the plane passing through P and having n as a normal. (a) P(-1,3,-2): n=(-2,1,-1) (b) P(1,1,4): n=(1,9,8) (c) P(2,0,0): n=(0,0,2) (d) P(0,0,0): n=(1,2,3)

Find the point-normal form of the equation of a plane given a point and a normal vector. This problem covers key concepts in analytic geometry, including how to derive the equation of a plane passing through a point with a given normal. Suitable for Grades 10-12 students.

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