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

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

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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