Math Problem Statement
Вычислить объем тетраэдра с вершинами в точках A(2, -3, 5), B(0, 2, 1), C(-2, -2, 3) и D(3, 2, 4).
Solution
To calculate the volume of a tetrahedron with vertices , , , and , we use the determinant formula:
x_B - x_A & x_C - x_A & x_D - x_A \\ y_B - y_A & y_C - y_A & y_D - y_A \\ z_B - z_A & z_C - z_A & z_D - z_A \end{bmatrix} \right|$$ ### Step 1: Define the vectors - $$\vec{AB} = B - A = (0 - 2, 2 - (-3), 1 - 5) = (-2, 5, -4)$$ - $$\vec{AC} = C - A = (-2 - 2, -2 - (-3), 3 - 5) = (-4, 1, -2)$$ - $$\vec{AD} = D - A = (3 - 2, 2 - (-3), 4 - 5) = (1, 5, -1)$$ ### Step 2: Construct the determinant The matrix is: $$\begin{bmatrix} -2 & -4 & 1 \\ 5 & 1 & 5 \\ -4 & -2 & -1 \end{bmatrix}$$ ### Step 3: Compute the determinant We expand the determinant as follows: $$\text{det} = -2 \cdot \begin{vmatrix} 1 & 5 \\ -2 & -1 \end{vmatrix} -4 \cdot \begin{vmatrix} 5 & 5 \\ -4 & -1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 5 & 1 \\ -4 & -2 \end{vmatrix}$$ #### Sub-determinants: 1. $$\begin{vmatrix} 1 & 5 \\ -2 & -1 \end{vmatrix} = (1)(-1) - (5)(-2) = -1 + 10 = 9$$ 2. $$\begin{vmatrix} 5 & 5 \\ -4 & -1 \end{vmatrix} = (5)(-1) - (5)(-4) = -5 + 20 = 15$$ 3. $$\begin{vmatrix} 5 & 1 \\ -4 & -2 \end{vmatrix} = (5)(-2) - (1)(-4) = -10 + 4 = -6$$ #### Substitute back: $$\text{det} = -2(9) - 4(15) + 1(-6) = -18 - 60 - 6 = -84$$ ### Step 4: Calculate the volume $$V = \frac{1}{6} \left| -84 \right| = \frac{84}{6} = 14$$ ### Final Answer: The volume of the tetrahedron is: $$\boxed{14 \, \text{units}^3}$$ --- Would you like further clarification or another example? Here are five related questions to explore further: 1. How is the determinant formula for tetrahedron volume derived? 2. Can the volume of a tetrahedron be calculated using the cross and dot product? 3. What happens to the volume if one vertex coincides with another? 4. How can the formula be generalized to higher dimensions (e.g., 4D simplex)? 5. What if the vertices were given in parametric form—how would the volume be computed? ### Tip: When working with determinants, simplify step-by-step to avoid sign errors and double-check sub-determinants.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Analytic Geometry
Vectors
Determinants
Volume of a Tetrahedron
Formulas
Volume of a Tetrahedron: V = (1/6) |det|
Determinant of a 3x3 matrix
Theorems
Determinant theorem for calculating the volume of a tetrahedron
Suitable Grade Level
Grades 11-12 / Advanced High School
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