Chapter 10: Q. 35 (page 824)

In Exercises 30–35 compute the indicated quantities when $u = (- 3, 1, - 4), v = (2, 0, 5), and w = (1, 3, 13)$
Find the volume of the parallelepiped determined by the vectors u, v, and w.

Short Answer

Expert verified

The volume of the parallelepiped determined by vectorsu,v andwis $0$ .

Step by step solution

Step 1. Given Information

The indicated quantities when $u = (- 3, 1, - 4), v = (2, 0, 5), and w = (1, 3, 13)$

We have to find the volume of the parallelepiped determined by the vectors u, v, andw.

Step 2. The volume of the parallelepiped determined by u, v, and w is the absolute value of the triple scalar product u·(v×w)

Although we could first evaluate the cross product $v \times w$ and then take the dot product of the resulting vector withu,it is slightly more efficient to just take the absolute value of the determinant of the 3 × 3 matrix formed from the components of u, v, and w as the rows.

Step 3. Thus, the required volume is

$u \cdot (v \times w) = \det [\begin{matrix} - 3 & 1 & - 4 \\ 2 & 0 & 5 \\ 1 & 3 & 13 \end{matrix}] u \cdot (v \times w) = - 3 |\begin{matrix} 0 & 5 \\ 3 & 13 \end{matrix}| - 1 |\begin{matrix} 2 & 5 \\ 1 & 13 \end{matrix}| - 4 |\begin{matrix} 2 & 0 \\ 1 & 3 \end{matrix}| u \cdot (v \times w) = - 3 (0 \times 13 - 5 \times 3) - 1 (2 \times 13 - 5 \times 1) - 4 (2 \times 3 - 0 \times 1) u \cdot (v \times w) = - 3 (0 - 15) - 1 (26 - 5) - 4 (6 - 0) u \cdot (v \times w) = 45 - 21 - 24 u \cdot (v \times w) = 0$

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In Exercises 30–35 compute the indicated quantities when $u = (- 3, 1, - 4), v = (2, 0, 5), and w = (1, 3, 13)$
Find the volume of the parallelepiped determined by the vectors u, v, and w.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The volume of the parallelepiped determined by u, v, and w is the absolute value of the triple scalar product u·(v×w)

Step 3. Thus, the required volume is

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In Exercises 30–35 compute the indicated quantities when u=(−3,1,−4),v=(2,0,5),andw=(1,3,13)Find the volume of the parallelepiped determined by the vectors u, v, and w.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The volume of the parallelepiped determined by u, v, and w is the absolute value of the triple scalar product u·(v×w)

Step 3. Thus, the required volume is

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Exercises 30–35 compute the indicated quantities when $u = (- 3, 1, - 4), v = (2, 0, 5), and w = (1, 3, 13)$
Find the volume of the parallelepiped determined by the vectors u, v, and w.