Nonstandard scalar product

Let the scalar product over \(\mathbb C^3\) be given as:

\(\langle \mathbf x|\mathbf y \rangle= x_1\overline{y_1} + x_2\overline{y_2} + 2x_3\overline{y_3} + x_3\overline{y_2} + x_2\overline{y_3} \)

Determine for the following vectors \(\mathbf x\) and \(\mathbf y\):

1. the scalar product of \(\mathbf x\) and \(\mathbf y\)

2. the Euclidean norms of \(\mathbf x\) and \(\mathbf y\)

3. the distance between \(\mathbf x\) and \(\mathbf y\)

4. whether vectors \(\mathbf x\) and \(\mathbf y\) are orthogonal.

Variant 1
\(\mathbf x^T=(4, 2, 3)\), \(\mathbf y^T=(1, 5, -2)\).
Result
\(\langle \mathbf x|\mathbf y \rangle =13\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=5\sqrt{2}\), \(||\mathbf y||=\sqrt{14}\), \(||\mathbf x-\mathbf y||=\sqrt{38}\).
Variant 2
\(\mathbf x^T=(3, 1, -2)\), \(\mathbf y^T=(1, -3, 2)\).
Result
\(\langle \mathbf x|\mathbf y \rangle =0\), \(\mathbf x\perp \mathbf y\), \(||\mathbf x||=\sqrt{14}\), \(||\mathbf y||=\sqrt{6}\), \(||\mathbf x-\mathbf y||=2\sqrt{5}\).
Variant 3
\(\mathbf x^T=(2, -1, 4)\), \(\mathbf y^T=(5, 2, -2)\).
Result
\(\langle \mathbf x|\mathbf y \rangle =2\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=\sqrt{29}\), \(||\mathbf y||=\sqrt{29}\), \(||\mathbf x-\mathbf y||=3\sqrt{6}\).
Variant 4
\(\mathbf x^T=(2+i, 0, 4-5i)\), \(\mathbf y^T=(1+i, 2+i, -1)\).
Result
\(\langle \mathbf x|\mathbf y \rangle =-2-5i\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=\sqrt{87}\), \(||\mathbf y||=\sqrt{5}\), \(||\mathbf x-\mathbf y||=4\sqrt{6}\).