Collection of Maths Problems

Product, norm, orthogonality

Task number: 2687

• Variant 1

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

• Hint

  \(||\mathbf x||=\sqrt{\langle \mathbf x|\mathbf x \rangle}\), and \(\mathbf x\perp \mathbf y\) when \(\langle \mathbf x|\mathbf y \rangle=0\).

• Result

  \(\langle \mathbf x|\mathbf y \rangle =8\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=\sqrt{29}\), \(||\mathbf y||=\sqrt{30}\), \(||\mathbf x-\mathbf y||=\sqrt{43}\),

• Variant 2

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

• Result

  \(\langle \mathbf x|\mathbf y \rangle =-4\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=\sqrt{14}\), \(||\mathbf y||=\sqrt{14}\), \(||\mathbf x-\mathbf y||=6\).

• Variant 3

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

• Result

  \(\langle \mathbf x|\mathbf y \rangle =0\), \(\mathbf x\perp \mathbf y\), \(||\mathbf x||=\sqrt{21}\), \(||\mathbf y||=\sqrt{33}\), \(||\mathbf x-\mathbf y||=3\sqrt{6}\),

• Variant 4

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

• Result

  \(\langle \mathbf x|\mathbf y \rangle =5\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=\sqrt{22}\), \(||\mathbf y||=\sqrt{21}\), \(||\mathbf x-\mathbf y||=\sqrt{33}\),

• Variant 5

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

• Resolution

  \(\mathbf x^T=(1, 1+i)\), \(\mathbf y^T=(2i, a+bi)\) hence \(\overline{\mathbf y}^T=(-2i, a-bi)\)

  \(\langle \mathbf x|\mathbf y \rangle = -2i + (1+i)(a-bi) = a+b + (a-b-2)i\)

  \(\mathbf x\perp \mathbf y\) for \(a=1\), \(b=-1\)

  \(||\mathbf x||=\sqrt{1^2+(1+i)(1-i)}=\sqrt{3}\)

  \(||\mathbf y||=\sqrt{2i(-2i)+(a+bi)(a-bi)}=\sqrt{a^2+b^2+4}\)

  \((\mathbf x-\mathbf y)^T=(1-2i,1-a+(1-b)i)\)

  \(||\mathbf x-\mathbf y||=\sqrt{a^2+b^2-2a-2b+7}\)

• Variant 6

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

• Result

  \(\langle \mathbf x|\mathbf y \rangle =4i-1\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=\sqrt{46}\), \(||\mathbf y||=2\sqrt{2}\), \(||\mathbf x-\mathbf y||=2\sqrt{14}\),

• Variant 7

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

• Result

  \(\langle \mathbf x|\mathbf y \rangle =-1-10i\), \(\mathbf x\not\perp \mathbf y\), \(||\mathbf x||=\sqrt{10}\), \(||\mathbf y||=\sqrt{11}\), \(||\mathbf x-\mathbf y||=\sqrt{23}\).