In general, what is the relationship between the volume of a parallelepiped spanned by three vectors $\vec{u}, \vec{v}, $ and $\vec{w},$ and the volume of the parallelepiped spanned by three vectors $\vec{u}, \vec{u}+\vec{v},$ and $\vec{u}+\vec{v}+\vec{w}?$ Make a reasonable guess and verify it algebraically, using properties of dot products and cross products.

Answered: In general, what is the relationship…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

In general, what is the relationship between the volume of a parallelepiped spanned by three vectors $\vec{u}, \vec{v}, $ and $\vec{w},$ and the volume of the parallelepiped spanned by three vectors $\vec{u}, \vec{u}+\vec{v},$ and $\vec{u}+\vec{v}+\vec{w}?$ Make a reasonable guess and verify it algebraically, using properties of dot products and cross products.

Expert Solution

Step 1: Introduction:

To find the relationship between the volume of a paralleloepiped spanned by three vectors $u with rightwards arrow on top comma v with rightwards arrow on top$ , and $w with rightwards arrow on top$ , and the volume of the parallelopiped spanned by three vectors $u with rightwards arrow on top comma u with rightwards arrow on top plus v with rightwards arrow on top$ , and $u with rightwards arrow on top plus v with rightwards arrow on top plus w with rightwards arrow on top$ .

Also, verify it algebraically using properties of dot products and cross products.

Concept used:

Volume of the parallelopiped is determined as either $open parentheses u cross times v close parentheses times w$ or $u times open parentheses v cross times w close parentheses$ .

Here, $u comma v comma w$ are the vectors in which the parallelopiped is formed by spanning of those vectors.

This triple product is also known as box product. it is notated as $open parentheses u cross times v close parentheses times w equals open square brackets u comma space v comma space w close square brackets$ .

The box product is nothing but the determinant of the matrix formed by vectors as each column of it.

So, if two vectors inside the box product are same, then the box product is zero.

$open square brackets a comma space b plus d comma space c close square brackets equals open square brackets a comma b comma c close square brackets plus open square brackets a comma d comma c close square brackets$ holds good as it is directly obtained from the determinant properties.