MathAlgebraLinear algebraVectors and vector spaces

Euclidean spaces

Theory

Orthogonal vectors

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The inner product of vectors v⃗1=(x1y1)T\vec{v}_{1} = \begin{pmatrix}x_{1}&y_{1}\end{pmatrix}^{\mathsf{T}} and v⃗2=(x2y2)T\vec{v}_{2} = {\begin{pmatrix}x_{2} & y_{2}\end{pmatrix}}^{\mathsf{T}} from R2\mathbb{R}^{2} is given by the following formula

⟨v⃗1,v⃗2⟩=2x1x2+3x1y2+3x2y1+6y1y2\left\lang\vec{v}_{1},\vec{v}_{2}\right\rang = 2x_{1}x_{2} + 3x_{1}y_{2} + 3x_{2}y_{1} + 6y_{1}y_{2}For example ⟨(11),(−13)⟩=2⋅1⋅(−1)+3⋅1⋅3+3⋅1⋅(−1)+6⋅1⋅3=−2+9−3+18=22\left\langle\begin{pmatrix}1\\1\end{pmatrix},\begin{pmatrix}-1\\3\end{pmatrix}\right\rangle = 2\cdot 1\cdot(-1) + 3\cdot 1\cdot 3 + 3\cdot 1\cdot(-1) + 6\cdot 1\cdot 3 = -2 + 9- 3+ 18 = 22Match the vectors from the following list that are orthogonal to each other.

Match the items from left and right columns
(10)T\begin{pmatrix}1&0\end{pmatrix}^{\mathsf{T}}
(−23)T\begin{pmatrix}-2&3\end{pmatrix}^{\mathsf{T}}
(−21)T\begin{pmatrix}-2&1\end{pmatrix}^{\mathsf{T}}
(01)T\begin{pmatrix}0&1\end{pmatrix}^{\mathsf{T}}
(−453)T\begin{pmatrix}-4&\frac{5}{3}\end{pmatrix}^{\mathsf{T}}
(−64)T\begin{pmatrix}-6&4\end{pmatrix}^{\mathsf{T}}
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