Math Problem Statement
find the angle between the two planes x + 6y − z = 6 and 6x − 6y + z = 1
Solution
To find the angle between two planes, we can use the formula:
Where:
- and are the normal vectors of the two planes.
- is the angle between the planes.
Step 1: Identify the normal vectors of the two planes
The equation of a plane is generally written in the form:
Where is the normal vector of the plane.
For the two planes given:
- has a normal vector
- has a normal vector
Step 2: Calculate the dot product of and
The dot product of two vectors and is given by:
Substitute the values from the normal vectors:
= 6 - 36 - 1 = -31$$ ### Step 3: Find the magnitudes of $$\mathbf{n_1}$$ and $$\mathbf{n_2}$$ The magnitude of a vector $$(a, b, c)$$ is: $$|\mathbf{n}| = \sqrt{a^2 + b^2 + c^2}$$ For $$\mathbf{n_1} = (1, 6, -1)$$: $$|\mathbf{n_1}| = \sqrt{1^2 + 6^2 + (-1)^2} = \sqrt{1 + 36 + 1} = \sqrt{38}$$ For $$\mathbf{n_2} = (6, -6, 1)$$: $$|\mathbf{n_2}| = \sqrt{6^2 + (-6)^2 + 1^2} = \sqrt{36 + 36 + 1} = \sqrt{73}$$ ### Step 4: Calculate $$\cos \theta$$ Now we can use the formula: $$\cos \theta = \frac{|\mathbf{n_1} \cdot \mathbf{n_2}|}{|\mathbf{n_1}| |\mathbf{n_2}|} = \frac{| -31 |}{\sqrt{38} \times \sqrt{73}} = \frac{31}{\sqrt{2774}}$$ $$\cos \theta = \frac{31}{52.66} \approx 0.5886$$ ### Step 5: Find the angle $$\theta$$ Now, take the inverse cosine to find the angle: $$\theta = \cos^{-1}(0.5886) \approx 54.00^\circ$$ ### Conclusion: The angle between the two planes is approximately $$54.00^\circ$$. Would you like more details on any step, or do you have questions? Here are some related problems: 1. How do you find the distance between two parallel planes? 2. What is the geometric interpretation of the dot product? 3. How can you find the line of intersection between two planes? 4. How do you find the angle between two vectors in 3D? 5. How do you find the distance between a point and a plane? **Tip**: The angle between two planes is the same as the angle between their normal vectors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Geometry
Dot Product
Angle Between Planes
Formulas
cos(θ) = |n1 • n2| / (|n1| |n2|)
Dot product: n1 • n2 = a1a2 + b1b2 + c1c2
Magnitude of a vector: |n| = √(a² + b² + c²)
Theorems
Dot Product Theorem
Plane Equation in 3D
Suitable Grade Level
Grades 10-12
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