Math Problem Statement

Regn ut volumet av parallelepipedet utspent av vektorene \( \mathbf{u} \), \( \mathbf{v} \) og \( \mathbf{w} \) hvor \( \mathbf{u} = [5, 1, 1] \), \( \mathbf{v} = [4, 8, 1] \), \( \mathbf{w} = [3, 2, 4] \).

Solution

To find the volume of a parallelepiped defined by three vectors u, v, and w, we can use the scalar triple product formula. Given:

[ \mathbf{u} = [5, 1, 1], \quad \mathbf{v} = [4, 8, 1], \quad \mathbf{w} = [3, 2, 4] ]

The volume VV of the parallelepiped is the absolute value of the scalar triple product:

V=∣u⋅(v×w)∣V = |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|

Step 1: Calculate the cross product v×w\mathbf{v} \times \mathbf{w}

Using the determinant method:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 8 & 1 \\ 3 & 2 & 4 \\ \end{vmatrix}$$ Expanding the determinant: $$= \mathbf{i} (8 \cdot 4 - 1 \cdot 2) - \mathbf{j} (4 \cdot 4 - 1 \cdot 3) + \mathbf{k} (4 \cdot 2 - 8 \cdot 3)$$ $$= \mathbf{i} (32 - 2) - \mathbf{j} (16 - 3) + \mathbf{k} (8 - 24)$$ $$= \mathbf{i} (30) - \mathbf{j} (13) + \mathbf{k} (-16)$$ So, $$\mathbf{v} \times \mathbf{w} = [30, -13, -16]$$ ### Step 2: Calculate the dot product $$\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$$ Now, we take the dot product of $$\mathbf{u} = [5, 1, 1]$$ with $$\mathbf{v} \times \mathbf{w} = [30, -13, -16]$$: $$\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = (5 \cdot 30) + (1 \cdot -13) + (1 \cdot -16)$$ $$= 150 - 13 - 16$$ $$= 121$$ ### Step 3: Take the absolute value The volume is the absolute value of this result: $$V = |121| = 121$$ ### Final Answer: The volume of the parallelepiped is $$\boxed{121}$$. Would you like more details or have any questions? --- 1. How is the scalar triple product defined geometrically? 2. What are the properties of the cross product that make it useful in volume calculations? 3. Why is the absolute value taken in the scalar triple product? 4. How would the volume change if one of the vectors were scaled? 5. What if two vectors were parallel—how would that affect the volume? **Tip:** Remember, the scalar triple product is zero if the vectors are coplanar (lying in the same plane), which would mean the parallelepiped has no volume.

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Math Problem Analysis

Mathematical Concepts

Vector Algebra
Scalar Triple Product
Cross Product

Formulas

Volume = |u · (v × w)|
Cross Product: \( v × w \) using determinant method
Dot Product: \( u · (v × w) \)

Theorems

Scalar Triple Product Theorem

Suitable Grade Level

Grades 10-12

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