S6.793.12.4.33 Find the volume of the parallelepiped

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In summary, the volume of the parallelepiped determined by the vectors $a, b,$ and $c$ is given by the magnitude of their scalar triple product, which can be calculated using the formula $V=|a\cdot(b\times c)|$. By plugging in the values for the given vectors, we get a value of $\color{red}{82}$ for the volume of the parallelepiped. The expand tab in the LaTeX window may not work on tablets, so we will have to be more concise in our posts.
  • #1
karush
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$\tiny{s6.793.12.4.33}$
$\textsf{
Find the volume of the parallelepiped determined by the vectors, a b and c}$
$ a =\langle 6, 3, -1\rangle
\, b =\langle 0, 1, 2 \rangle
\, c =\langle 4, -2, 5 \rangle $

$\textsf{The volumn of the parallelepiped determined by the vectors }\\$
$\textsf{ $a, b$ and $c$ is the magnitude of their scalar triple product.}$

\begin{align}
\displaystyle
V&=|a \cdot(b \times c)|\\
\end{align}
then
\begin{align}
V=|a \cdot(b \times c)|&=
\begin{bmatrix}
6 & 3 & -1\\
0 &1 &2\\
4 &-2 &5
\end{bmatrix} \\
&=6\begin{bmatrix}=
1 &2\\
-2 &5
\end{bmatrix}
+3\begin{bmatrix}
0 &2\\
4 &5
\end{bmatrix}
-\begin{bmatrix}
0 &1\\
4 &-2
\end{bmatrix}\\
&= 6(9)-3(-8) +(4) \\
&\color{red}{V=82}
\end{align}
$\textit{ok think this is ok. but always suggestions! }\\$
$\textit{btw need more lines to expand to in latex window scrolling constantly not fun}$😰
 
Last edited:
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  • #2
Looks good to me. A few errors in transferring your notes to here, but i think you have understood the process.

btw need more lines to expand to in latex window scrolling constantly not funbtw need more lines to expand to in latex window scrolling constantly not fun

I agree! There is a little tab down the bottom right of the text window that you can pull to increase the size.

EDIT: Ohh that window... I didn't even know it was there :). I usually just type in the big window and then hit preview to check my LaTeX.
 
  • #3
\begin{align}
V=|a \cdot(b \times c)|&=
\begin{bmatrix}
6 & 3 & -1\\
0 &1 &2\\
4 &-2 &5
\end{bmatrix} \\
&=6\begin{bmatrix}
1 &2\\
-2 &5
\end{bmatrix}
+3\begin{bmatrix}
0 &2\\
4 &5
\end{bmatrix}
-\begin{bmatrix}
0 &1\\
4 &-2
\end{bmatrix}\\
&=6(9)-3(-8) +(4) \\
&=\color{red}{82}
\end{align}
$\textit{think this is it. well the expand tab doesn't work on a tablet! }$
 
  • #4
Looks better :)

karush said:
$\textit{think this is it. well the expand tab doesn't work on a tablet! }$

True.. I just tried on my iPad and it doesn't work. I guess we'll just have to be more concise with our posts xD.
 

Related to S6.793.12.4.33 Find the volume of the parallelepiped

1. What is a parallelepiped?

A parallelepiped is a 3-dimensional geometric shape that is formed by six parallelograms. It has 6 faces, 12 edges, and 8 vertices.

2. How is the volume of a parallelepiped calculated?

The volume of a parallelepiped can be calculated by multiplying the lengths of its three sides, also known as its base, by its height. The formula for calculating the volume is V = l x w x h, where l is the length, w is the width, and h is the height.

3. What are the units of measurement for the volume of a parallelepiped?

The units of measurement for the volume of a parallelepiped will depend on the units used for its length, width, and height. For example, if the lengths are measured in meters, then the volume will be in cubic meters (m3).

4. How do I find the length, width, and height of a parallelepiped?

To find the length, width, and height of a parallelepiped, you will need to measure the three sides using a ruler or measuring tape. If the parallelepiped is not a regular shape, you may need to use a formula or calculate the measurements from known angles and sides.

5. Can the volume of a parallelepiped be negative?

No, the volume of a parallelepiped cannot be negative. Volume is a measure of space, and it cannot have a negative value. If the measurements used to calculate the volume are negative, then the result will also be negative.

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