Math Problem Statement
Solution
To solve the problem of finding the distance from the point to the plane , we use the formula for the distance between a point and a plane :
Step 1: Rewrite the plane equation
The given equation is: Rewrite it in standard form : Here, , , , and .
Step 2: Substitute the point into the formula
The coordinates of the point are , so , , and . Plugging these into the distance formula:
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$$.Ask a new question for Free
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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
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