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We are given a plane equation:

\[
3x - 4z + 15 = 0
\]

The task is to find the coordinates of the normal vector to this plane, where the length of the vector is 1 and its first component is positive.

### Step 1: Find the normal vector to the plane

The general form of a plane equation is:

\[
Ax + By + Cz + D = 0
\]

The normal vector to the plane is given by the coefficients of \(x\), \(y\), and \(z\), which in this case are \(A = 3\), \(B = 0\) (since there is no \(y\)-term), and \(C = -4\).

Thus, the normal vector is:

\[
\mathbf{n} = (3, 0, -4)
\]

### Step 2: Normalize the vector

To find the unit vector, we need to normalize the normal vector. The magnitude of the vector \(\mathbf{n} = (3, 0, -4)\) is:

\[
|\mathbf{n}| = \sqrt{3^2 + 0^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Now, divide each component of the vector by 5 to normalize it:

\[
\mathbf{n_{\text{unit}}} = \left( \frac{3}{5}, 0, \frac{-4}{5} \right)
\]

### Step 3: Ensure the first component is positive

The first component \( \frac{3}{5} \) is already positive, so we do not need to change the direction of the vector.

### Step 4: Answer the question

The first coordinate of the unit normal vector is \( \frac{3}{5} \).

---

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the equation of a plane given a normal vector and a point?
2. What is the geometric significance of the normal vector to a plane?
3. How do you compute the angle between two planes using their normal vectors?
4. What is the relationship between the gradient of a surface and the normal vector?
5. How do you verify if a point lies on a given plane?

**Tip:** Always check the direction of your normal vector to ensure it satisfies any specific conditions in the problem, like positivity of certain components.

For the plane defined by the equation 3x - 4z + 15 = 0, find the coordinates of the normal vector. Provide the vector with length 1, where the first component is positive. What is the first coordinate of this vector?

Explore how to calculate and normalize the normal vector for the plane equation 3x - 4z + 15 = 0. This guide covers the principles of vector algebra and normalization, suitable for college-level mathematics.

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