Find an equation of the plane.
the plane through the point 
(4, −3, 5)
 and perpendicular to the vector 
−i + 3j + 4k
 

The equation of a plane in 3D space is given by:

\[
Ax + By + Cz = D
\]

Where \( \mathbf{n} = (A, B, C) \) is the normal vector to the plane and \( (x_0, y_0, z_0) \) is a point on the plane.

We are given:
- A point on the plane: \( (4, -3, 5) \)
- A normal vector: \( \mathbf{n} = -\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \), which corresponds to the components \( (-1, 3, 4) \).

The general equation of the plane is:
\[
A(x - x_0) + B(y - y_0) + C(z - z_0) = 0
\]

Substituting the point \( (x_0, y_0, z_0) = (4, -3, 5) \) and the normal vector \( (-1, 3, 4) \), we get:

\[
-1(x - 4) + 3(y + 3) + 4(z - 5) = 0
\]

Simplifying:

\[
-(x - 4) + 3(y + 3) + 4(z - 5) = 0
\]

Distribute:

\[
-x + 4 + 3y + 9 + 4z - 20 = 0
\]

\[
-x + 3y + 4z - 7 = 0
\]

Rearranging the terms:

\[
x - 3y - 4z = -7
\]

Thus, the equation of the plane is:

\[
x - 3y - 4z = -7
\]

Let me know if you would like more details or have any questions.

Here are 5 related questions for practice:
1. Find the equation of a plane passing through \( (1, 2, 3) \) and perpendicular to \( 2\mathbf{i} - \mathbf{j} + \mathbf{k} \).
2. Determine the distance from the point \( (5, 2, 6) \) to the plane \( 2x - 3y + 4z = 10 \).
3. What is the angle between the planes \( x + y + z = 0 \) and \( 2x - 3y + z = 5 \)?
4. Find the equation of the plane through the points \( (1, 0, 0) \), \( (0, 1, 0) \), and \( (0, 0, 1) \).
5. If a plane passes through the origin and is perpendicular to the vector \( \mathbf{i} + \mathbf{j} + \mathbf{k} \), what is its equation?

**Tip:** When a plane is perpendicular to a vector, that vector becomes the normal vector of the plane.

Find an equation of the plane. The plane through the point (4, −3, 5) and perpendicular to the vector −i + 3j + 4k.

This page provides a step-by-step solution to finding the equation of a plane through the point (4, -3, 5) and perpendicular to the vector -i + 3j + 4k. Learn about vectors, planes in 3D space, and key geometric concepts. Suitable for Grades 11-12 or early college students.

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