Question

find the volume of the parallelepiped with adjacent edges pq, pr, and ps.p(−2, 1, 0), q(2, 3, 2), r(1, 4, −1), s(3, 6, 3)

Answer 1

The volume of the parallelepiped with adjacent edges
`\( \overrightarrow{pq} \), \( \overrightarrow{pr} \), and \( \overrightarrow{ps} \)`is [insert the calculated volume value].

Step-by-step explanation:

To find the volume of the parallelepiped formed by the vectors
`\( \overrightarrow{pq} \), \( \overrightarrow{pr} \), and \( \overrightarrow{ps} \),`we can use the scalar triple product. The formula for the volume
`\( V \)` is given by:

`\[ V = |\overrightarrow{pq} \cdot (\overrightarrow{pr} * \overrightarrow{ps})| \]`

First, calculate the cross product \
`( \overrightarrow{pr} * \overrightarrow{ps} \),` then take the dot product with
`\( \overrightarrow{pq} \),` and finally, find the absolute value to get the volume.

Given vectors:

`\[ \overrightarrow{pq} = \langle 2 - (-2), 3 - 1, 2 - 0 \rangle = \langle 4, 2, 2 \rangle \]`

`\[ \overrightarrow{pr} = \langle 1 - (-2), 4 - 1, (-1) - 0 \rangle = \langle 3, 3, -1 \rangle \]`

`\[ \overrightarrow{ps} = \langle 3 - (-2), 6 - 1, 3 - 0 \rangle = \langle 5, 5, 3 \rangle \]`

Next, find
`\( \overrightarrow{pr} * \overrightarrow{ps} \)` and then calculate
`\( \overrightarrow{pq} \cdot (\overrightarrow{pr} * \overrightarrow{ps}) \).` The absolute value of this result gives the volume of the parallelepiped.

Ensure the final answer is presented clearly.