Math Problem Statement

Find the distance from the point (-3, 1, 0) to the plane -3x + 3y + 3z = 9.

Solution

To solve the problem of finding the distance from the point (3,1,0)(-3, 1, 0) to the plane 3x+3y+3z=9-3x + 3y + 3z = 9, we use the formula for the distance between a point (x1,y1,z1)(x_1, y_1, z_1) and a plane Ax+By+Cz+D=0Ax + By + Cz + D = 0:

Distance=Ax1+By1+Cz1+DA2+B2+C2\text{Distance} = \frac{|A x_1 + B y_1 + C z_1 + D|}{\sqrt{A^2 + B^2 + C^2}}

Step 1: Rewrite the plane equation

The given equation is: 3x+3y+3z=9-3x + 3y + 3z = 9 Rewrite it in standard form Ax+By+Cz+D=0Ax + By + Cz + D = 0: 3x+3y+3z9=0-3x + 3y + 3z - 9 = 0 Here, A=3A = -3, B=3B = 3, C=3C = 3, and D=9D = -9.

Step 2: Substitute the point into the formula

The coordinates of the point are (3,1,0)(-3, 1, 0), so x1=3x_1 = -3, y1=1y_1 = 1, and z1=0z_1 = 0. Plugging these into the distance formula:

Distance=(3)(3)+(3)(1)+(3)(0)9(3)2+32+32\text{Distance} = \frac{|(-3)(-3) + (3)(1) + (3)(0) - 9|}{\sqrt{(-3)^2 + 3^2 + 3^2}}

Step 3: Simplify the numerator

= |9 + 3 + 0 - 9| = |3|$$ ### Step 4: Simplify the denominator $$\text{Denominator} = \sqrt{(-3)^2 + 3^2 + 3^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3}$$ ### Step 5: Compute the distance $$\text{Distance} = \frac{3}{3\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$ ### Final Answer: The distance from the point $$(-3, 1, 0)$$ to the plane $$-3x + 3y + 3z = 9$$ is: $$\boxed{\frac{\sqrt{3}}{3}}$$ Would you like further clarification or steps explained in detail? --- Here are 5 related questions to explore: 1. What is the geometric meaning of the distance between a point and a plane? 2. How do you find the equation of a plane given three points? 3. Can the distance formula be applied in higher dimensions (e.g., 4D space)? 4. How is the distance formula derived from vector projections? 5. What changes if the plane equation is written differently (e.g., with coefficients scaled)? ### Tip: To simplify calculations, always ensure the plane equation is written in standard form $$Ax + By + Cz + D = 0$$.

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Math Problem Analysis

Mathematical Concepts

Geometry
Distance from a point to a plane
Vectors

Formulas

Distance = |Ax1 + By1 + Cz1 + D| / sqrt(A^2 + B^2 + C^2)

Theorems

Distance formula for a point and a plane

Suitable Grade Level

Grades 10-12