Find the distance from the point (2, 5, 3) to the plane −5x−y−4z=4.

The formula for the distance from a point \((x_1, y_1, z_1)\) to a plane given by the equation \(Ax + By + Cz + D = 0\) is:

\[
\text{Distance} = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}
\]

The equation of the plane in the problem is:

\[
-5x - y - 4z = 4
\]

We can rewrite this as:

\[
-5x - y - 4z - 4 = 0
\]

Here, \(A = -5\), \(B = -1\), \(C = -4\), and \(D = -4\).

The point given is \((2, 5, 3)\), so \(x_1 = 2\), \(y_1 = 5\), and \(z_1 = 3\).

Now, substitute the values into the distance formula:

\[
\text{Distance} = \frac{|-5(2) - 1(5) - 4(3) - 4|}{\sqrt{(-5)^2 + (-1)^2 + (-4)^2}}
\]

Let's calculate the result step by step.

### Numerator:

\[
|-10 - 5 - 12 - 4| = |-31| = 31
\]

### Denominator:

\[
\sqrt{(-5)^2 + (-1)^2 + (-4)^2} = \sqrt{25 + 1 + 16} = \sqrt{42}
\]

Thus, the distance is:

\[
\text{Distance} = \frac{31}{\sqrt{42}} \approx 4.78
\]

The distance from the point \((2, 5, 3)\) to the plane \(-5x - y - 4z = 4\) is approximately \(4.78\) units.

Would you like further details, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. How would the formula change for the distance between two points in 3D space?
2. What is the geometric interpretation of a plane in three-dimensional space?
3. Can you derive the formula for the distance from a point to a plane?
4. How does the distance from a point to a plane relate to the plane's normal vector?
5. How would you find the equation of a plane passing through three points?

**Tip:** In three-dimensional geometry, the normal vector of a plane (derived from the coefficients \(A\), \(B\), and \(C\)) plays a key role in determining distances and angles relative to the plane.

Find the distance from the point (2, 5, 3) to the plane −5x − y − 4z = 4.

Calculate the distance from the point (2, 5, 3) to the plane −5x − y − 4z = 4 using the formula for the distance between a point and a plane. This problem explores key geometric and algebraic concepts in analytic geometry. Ideal for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation