Orthogonale Vektore aufstellen

Question

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Problem 1 Orthogonal vectors
(a) Consider the vector \( \mathbf{v}:=\left(\begin{array}{l}3 \\ 1\end{array}\right) \in \mathbb{R}^{2} \). Find a vector \( \mathbf{u} \in \mathbb{R}^{2} \) such that \( \mathbf{u} \perp \mathbf{v} \) and \( \|\mathbf{u}\|=3\|\mathbf{v}\| \).
\( \begin{array}{l} v \perp \mu=\left(\begin{array}{l} 3 \\ 1 \end{array}\right) \perp\left(\begin{array}{c} -1 \\ 3 \end{array}\right) \\ \|V\|=\sqrt{3^{2}+1^{2}}=\sqrt{10} \\ \|\mu\|=3 \cdot \sqrt{10}=3 \sqrt{10} \end{array} \)
(b) Find two vectors \( \mathbf{r} \in \mathbb{R}^{3} \) and \( \mathbf{s} \in \mathbb{R}^{3} \) different from the zero vector \( \mathbf{o} \in \mathbb{R}^{3} \), which are orthogonal to the vector \( \mathbf{t}:=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) \in \mathbb{R}^{3} \) and orthogonal to each other.

Aufgabe:

Problem/Ansatz: ich verstehe das einfach nicht

Answer 1

a)

v soll dreimal so lang sein wie u. Damit brauchst du nur mit 3 oder -3 multiplizieren.

3 * [-1, 3] = [-3, 9]

-3 * [-1, 3] = [3, -9]

b)

r = [1, -1, 0]

s = [1, 1, 1] ⨯ [1, -1, 0] = [1, 1, -2]