Question

Find the distance from (3,-1,2) to the plane \(5 x-y-z=4\).

Short answer

The distance is \( \frac{10\sqrt{3}}{9} \).

Step-by-step solution

- Understand the formula for distance

The distance from a point \((x_1, y_1, z_1)\) to the plane \(Ax + By + Cz + D = 0\) is given by the formula: \[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

- Identify the coefficients and point coordinates

For the given plane \(5x - y - z = 4\), rewrite it in standard form: \(5x - y - z - 4 = 0\). Here, \(A = 5\), \(B = -1\), \(C = -1\), and \(D = -4\). The point given is \( (3, -1, 2)\), so \(x_1 = 3\), \(y_1 = -1\), and \(z_1 = 2\).

- Substitute the values into the formula

Substitute the values into the distance formula: \[ d = \frac{|5(3) - 1(-1) - 1(2) - 4|}{\sqrt{5^2 + (-1)^2 + (-1)^2}} \]

- Simplify the numerator

Calculate the numerator: \5(3) = 15\, \-1(-1) = 1\, \-1(2) = -2\. So, the numerator becomes: \[ |15 + 1 - 2 - 4| = |10| = 10 \]

- Simplify the denominator

Calculate the denominator: \[ \sqrt{5^2 + (-1)^2 + (-1)^2} = \sqrt{25 + 1 + 1} = \sqrt{27} = 3\sqrt{3} \]

- Compute the distance

Combine the simplified numerator and denominator to find the distance: \[ d = \frac{10}{3\sqrt{3}} \]. Simplify this result to get the final distance: \[ d = \frac{10\sqrt{3}}{9} \].

Key concepts

distance formula

The distance formula is a key tool in coordinate geometry. It helps find the straight-line distance between a point and various geometric objects like lines or planes. When calculating the distance from a point \((x_1, y_1, z_1)\) to a plane given by \(Ax + By + Cz + D = 0\), use the formula:
\[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\text{√}(A^2 + B^2 + C^2)} \].
This formula essentially leverages the perpendicular distance from the point to the plane. Here's a quick breakdown:

• The numerator \(|Ax_1 + By_1 + Cz_1 + D|\) represents the absolute value of the plane equation evaluated at the point \((x_1, y_1, z_1)\).
• The denominator \( \text{√}(A^2 + B^2 + C^2) \) normalizes the vector coefficients of the plane.

This approach ensures that the distance is the shortest possible from the point to the plane, making it extremely convenient for geometric calculations and real-world applications.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinates to define and study geometric shapes and properties. In this exercise, the coordinate geometry focuses on calculating the distance between a point and a plane in a three-dimensional space.
We can think of coordinates as a way to pinpoint exact locations in space using an ordered set of numbers, typically \(x, y, z\). In our exercise, the plane equation and the point coordinates were given:

• The plane: \(5x - y - z = 4\)
• The point: \((3, -1, 2)\)

By placing these coordinates into a structured formula, coordinate geometry helps bridge algebra and geometry, making complex shapes easier to understand and analyze.

plane equation

A plane equation in three-dimensional space can generally be written as \(Ax + By + Cz + D = 0\). This represents a flat surface extending infinitely in all directions within the 3D space.
The coefficients \(A, B, C\) represent the normal vector to the plane, which is perpendicular to the plane's surface. The term \(D\) shifts the plane from the origin. In this example:

• The original plane equation was given as \(5x - y - z = 4\).
• We rewrite it to standard form: \(5x - y - z - 4 = 0\), where \(A = 5\), \(B = -1\), \(C = -1\), and \(D = -4\).

Understanding the plane equation is crucial because it allows us to apply formulas to calculate various geometric properties, like distance from a point or the angle between two planes. It also serves as a basis for solving more advanced problems in physics and engineering.