## 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

$$\mathbf x^T=(4, 2, 3)$$, $$\mathbf y^T=(1, 5, -2)$$.

• #### Variant 2

$$\mathbf x^T=(3, 1, -2)$$, $$\mathbf y^T=(1, -3, 2)$$.

• #### Variant 3

$$\mathbf x^T=(2, -1, 4)$$, $$\mathbf y^T=(5, 2, -2)$$.

• #### Variant 4

$$\mathbf x^T=(2+i, 0, 4-5i)$$, $$\mathbf y^T=(1+i, 2+i, -1)$$.