Math Problem Statement

Find the volume of the parallelepiped that has u = ⟨2, -8, 2⟩, v = ⟨0, 4, -2⟩, and w = ⟨1, 2, -4⟩ as the adjacent sides.

Solution

To find the volume of the parallelepiped defined by the vectors u=⟨2,−8,2⟩\mathbf{u} = \langle 2, -8, 2 \rangle, v=⟨0,4,−2⟩\mathbf{v} = \langle 0, 4, -2 \rangle, and w=⟨1,2,−4⟩\mathbf{w} = \langle 1, 2, -4 \rangle, we need to compute the scalar triple product u⋅(v×w)\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).

Step 1: Compute v×w\mathbf{v} \times \mathbf{w} (cross product of v\mathbf{v} and w\mathbf{w}).

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 4 & -2 \\ 1 & 2 & -4 \end{vmatrix}$$ This determinant expands as follows: $$\mathbf{v} \times \mathbf{w} = \mathbf{i} \left( 4(-4) - (-2)(2) \right) - \mathbf{j} \left( 0(-4) - (-2)(1) \right) + \mathbf{k} \left( 0(2) - 4(1) \right)$$ Simplifying each term: $$\mathbf{i}( -16 + 4 ) - \mathbf{j}( 0 + 2 ) + \mathbf{k}( 0 - 4 )$$ $$= \mathbf{i}(-12) - \mathbf{j}(2) + \mathbf{k}(-4)$$ Thus, $$\mathbf{v} \times \mathbf{w} = \langle -12, -2, -4 \rangle$$. ### Step 2: Compute $$\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$$ (dot product of $$\mathbf{u}$$ and the result from Step 1). $$\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \langle 2, -8, 2 \rangle \cdot \langle -12, -2, -4 \rangle$$ The dot product is calculated as: $$2(-12) + (-8)(-2) + 2(-4) = -24 + 16 - 8 = -16$$ ### Step 3: Take the absolute value of the result. The volume of the parallelepiped is the absolute value of the scalar triple product: $$\text{Volume} = | -16 | = 16$$ Thus, the volume of the parallelepiped is **16 cubic units**. --- Do you have any questions or need further clarification? Here are some related questions for further exploration: 1. What is the geometric meaning of the cross product in the context of parallelepipeds? 2. How do you compute the cross product of two 3D vectors step-by-step? 3. What would happen to the volume if one of the vectors is a scalar multiple of another? 4. How can the scalar triple product help determine if three vectors are coplanar? 5. How would you find the area of the parallelogram formed by two vectors in 3D? **Tip:** The scalar triple product's result being zero means the vectors are coplanar, implying no volume.

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Math Problem Analysis

Mathematical Concepts

Vector Operations
Scalar Triple Product
Cross Product
Dot Product

Formulas

Scalar Triple Product: u ⋅ (v × w)
Cross Product Formula: v × w = determinant of a 3x3 matrix
Dot Product Formula: u ⋅ (v × w)

Theorems

Vector Cross Product
Scalar Triple Product Theorem

Suitable Grade Level

Grades 11-12 or College level

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