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Linear Algebra

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Inner Product Space: Orthogonal Vector in Inner Product Space

LINALG-KOG97V

Define an inner product on $\mathbb{R}^3$ by the rule if $\boldsymbol{u}=\begin{bmatrix} u_1 \\\ u_2 \\\ u_3 \end{bmatrix},\ \boldsymbol{v}=\begin{bmatrix} v_1 \\\ v_2 \\\ v_3\end{bmatrix} \in \mathbb{R}^3$ then:

$$\langle \boldsymbol{u},\ \boldsymbol{v}\rangle = u_1v_1 + \dfrac{1}{2}u_2v_2 + \dfrac{1}{4}u_3v_3$$

Determine which of the following vectors are orthogonal to $\boldsymbol{u} = \begin{bmatrix} -1 \\\ 2 \\\ 1\end{bmatrix}$ with respect to the inner product defined above.

Select ALL that apply.

A

$\begin{bmatrix} 0 \\\ 1 \\\ 1\end{bmatrix}$

B

$\begin{bmatrix} 5/4 \\\ 1 \\\ 1\end{bmatrix}$

C

$\begin{bmatrix} 3 \\\ 1 \\\ 1\end{bmatrix}$

D

$\begin{bmatrix} 1 \\\ 0 \\\ 1\end{bmatrix}$

E

$\begin{bmatrix} -1 \\\ 1 \\\ -8\end{bmatrix}$

F

$\begin{bmatrix} 0 \\\ 0 \\\ 0\end{bmatrix}$