Question

For given vectors: P1 = (1, 2,−1, 0), P2 = (0, 2, 1, 1), P3 = (1,−1, 1, 3), P4 = (1, 0, 0, 1) • find the vector orthogonal to vectors P2,P3, and P4 • volume of the parallelepiped spanned by the given vectors.

Answer 1

a) To find the vector that is orthogonal to vectors P2, P3, and P4, start by finding the vector equation for each of the given vectors.

P1 = (1, 2,−1, 0) →

P2 = (0, 2, 1, 1) →

P3 = (1,−1, 1, 3) →

P4 = (1, 0, 0, 1)

Then, the vector that is orthogonal to these given vectors is:
(0, -2, 0, 0)

Answer 2

The vector orthogonal to vectors
`\( \mathbf{P}_2 \)`,
`\( \mathbf{P}_3 \)`, and
`\( \mathbf{P}_4 \)` is
`\( \mathbf{v} = (5, -3, -2, -4) \)`. The volume of the parallelepiped spanned by the given vectors is
`\( V = 15 \)`.

Step-by-step explanation:

To find the vector orthogonal to
`\( \mathbf{P}_2 \)`,
`\( \mathbf{P}_3 \)`, and
`\( \mathbf{P}_4 \`), we can use the cross product. Let
`\( \mathbf{v} \)` be the vector we are looking for. The cross product of
`\( \mathbf{P}_2 \)`,
`\( \mathbf{P}_3 \)`, and
`\( \mathbf{P}_4 \)` is given by:

`\[ \mathbf{v} = \mathbf{P}_2 * \mathbf{P}_3 * \mathbf{P}_4 \]`

Substitute the given values, perform the cross product, and we get
`\( \mathbf{v} = (5, -3, -2, -4) \)`.

Now, to find the volume of the parallelepiped spanned by the vectors
`\( \mathbf{P}_1 \)`,
`\( \mathbf{P}_2 \)`, and
`\( \mathbf{P}_3 \)`, we can use the scalar triple product. The volume
`\( V \)` is given by:

`\[ V = \left| \mathbf{P}_1 \cdot (\mathbf{P}_2 * \mathbf{P}_3) \right| \]`

Substitute the given values, perform the calculations, and we find
`\( V = 15 \)`. The absolute value is taken to ensure a positive volume.

In conclusion, the orthogonal vector
`\( \mathbf{v} \)` is found using the cross product, and the volume
`\( V \)` of the parallelepiped is determined using the scalar triple product.