?

Linear Algebra

Free Version

Upgrade subject to access all content

Easy

Inner Product Space: Geometric Inner Product in $R^3$

LINALG-9DJEYW

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:

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

Compute $\langle \boldsymbol{u},\ \boldsymbol{v}\rangle$ if $\boldsymbol{u}=\begin{bmatrix} 1 \\\ 0 \\\ 2 \end{bmatrix}$ and $\boldsymbol{v}=\begin{bmatrix} 3 \\\ 7 \\\ -2 \end{bmatrix}$.

A

$0$

B

$1$

C

$2$

D

$3$

E

$4$

F

$5$