## Product, norm, orthogonality

### Task number: 2687

For the standard scalar product $$\langle \mathbf x|\mathbf y \rangle=\sum_{i=1}^n x_i \overline{y_i}$$ over $$\mathbb C^n$$, or $$\mathbb R^n$$ 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

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

• #### Variant 5

$$\mathbf x^T=(1, 1+i)$$, $$\mathbf y^T=(2i, a+bi)$$ (with real parameters $$a$$, $$b$$)

• #### Variant 6

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

• #### Variant 7

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