How do you find the volume of the parallelepiped determined by the vectors: , and ?

Question

How do you find the volume of the parallelepiped determined by the vectors: <1,3,7>, <2,1,5> and <3,1,1>?

Elijah Abram · Accepted Answer

I would use the Triple Scalar Product (or Box Product):

Luke Alvarez · Answer

undefined To find the volume of the parallelepiped determined by the vectors $ \langle 1, 3, 7 \rangle $, $ \langle 2, 1, 5 \rangle $, and $ \langle 3, 1, 1 \rangle $, you can use the scalar triple product. The absolute value of the scalar triple product of these vectors gives the volume of the parallelepiped.

The formula for the scalar triple product is:

$$ V = |(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}))| $$

where $ \mathbf{a} $, $ \mathbf{b} $, and $ \mathbf{c} $ are the given vectors.

So, calculate the cross product of $ \langle 2, 1, 5 \rangle $ and $ \langle 3, 1, 1 \rangle $, then take the dot product of the result with $ \langle 1, 3, 7 \rangle $, and finally take the absolute value of the result.

$$ \langle 2, 1, 5 \rangle \times \langle 3, 1, 1 \rangle = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 1 & 5 \ 3 & 1 & 1 \end{vmatrix} = \langle 24, 13, -5 \rangle $$

Now, take the dot product of $ \langle 1, 3, 7 \rangle $ and $ \langle 24, 13, -5 \rangle $:

$$ \langle 1, 3, 7 \rangle \cdot \langle 24, 13, -5 \rangle = (1)(24) + (3)(13) + (7)(-5) = 24 + 39 - 35 = 28 $$

Finally, take the absolute value of the result:

$$ |28| = 28 $$

So, the volume of the parallelepiped determined by the given vectors is $ 28 $ cubic units.

How do you find the volume of the parallelepiped determined by the vectors: &lt;1,3,7&gt;, &lt;2,1,5&gt; and &lt;3,1,1&gt;?

How do you find the volume of the parallelepiped determined by the vectors: <1,3,7>, <2,1,5> and <3,1,1>?