Question

find the volume of the parallelepiped with adjacent edges pq, pr, ps. p(1, 0, 3), q(−3, 1, 7), r(4, 2, 2), s(0, 6, 5)

Answer 1

The volume of the parallelepiped with adjacent edges PQ, PR, PS, where P(1, 0, 3), Q(-3, 1, 7), R(4, 2, 2), S(0, 6, 5), is 143 cubic units.

To find the volume
`\( V \)` of a parallelepiped given three vectors
`\( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \)`representing the adjacent edges, you can use the following formula:

`\[ V = |\mathbf{u} \cdot (\mathbf{v} * \mathbf{w})| \]`

Here,
`\( * \)` denotes the cross product, and
`\( \cdot \)` denotes the dot product.

Let's denote the vectors
`\( \mathbf{pq} \), \( \mathbf{pr} \), and \( \mathbf{ps} \) as \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \)`respectively.

`\[ \mathbf{u} = \mathbf{q} - \mathbf{p} \]`

`\[ \mathbf{v} = \mathbf{r} - \mathbf{p} \]`

`\[ \mathbf{w} = \mathbf{s} - \mathbf{p} \]`

Let's calculate
`\( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \)` first:

`\[ \mathbf{u} = \mathbf{q} - \mathbf{p} = \langle -3 - 1, 1 - 0, 7 - 3 \rangle = \langle -4, 1, 4 \rangle \]`

`\[ \mathbf{v} = \mathbf{r} - \mathbf{p} = \langle 4 - 1, 2 - 0, 2 - 3 \rangle = \langle 3, 2, -1 \rangle \]`

`\[ \mathbf{w} = \mathbf{s} - \mathbf{p} = \langle 0 - 1, 6 - 0, 5 - 3 \rangle = \langle -1, 6, 2 \rangle \]`

Now, compute
`\( \mathbf{v} * \mathbf{w} \)` (the cross product):

`\[ \mathbf{v} * \mathbf{w} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 2 & -1 \\ -1 & 6 & 2 \end{vmatrix} \]`

`\[ \mathbf{v} * \mathbf{w} = \langle -14, -7, 20 \rangle \]`

Finally, calculate
`\( \mathbf{u} \cdot (\mathbf{v} * \mathbf{w}) \)` (the dot product):

`\[ \mathbf{u} \cdot (\mathbf{v} * \mathbf{w}) = \langle -4, 1, 4 \rangle \cdot \langle -14, -7, 20 \rangle \]`

`\[ \mathbf{u} \cdot (\mathbf{v} * \mathbf{w}) = (-4)(-14) + (1)(-7) + (4)(20) \]`

`\[ \mathbf{u} \cdot (\mathbf{v} * \mathbf{w}) = 56 + 7 + 80 \]`

`\[ \mathbf{u} \cdot (\mathbf{v} * \mathbf{w}) = 143 \]`

The volume \( V \) is the absolute value of this result:

`\[ V = |143| = 143 \]`

So, the volume of the parallelepiped is 143 cubic units.