​ Vector Problem

Vector Problem

$\begin{array}{|rcll|} \hline && |3\vec{x}+2\vec{y}| \qquad \text{Formula: $|\vec{v}|=\sqrt{\vec{v}\cdot \vec{v}}$ } \\\\ &=& \sqrt{ (3\vec{x}+2\vec{y}) (3\vec{x}+2\vec{y}) } \\ &=& \sqrt{ 9\vec{x}^2 + 4 \vec{y}^2 + 2\cdot 3 \cdot 2\cdot( \vec{x}\cdot \vec{y}) } \\\\ && \qquad \text{Formula: $|\vec{x}^2|=|\vec{x}|^2 = x^2 \qquad |\vec{y}^2|=|\vec{y}|^2 = y^2$ } \\\\ &=& \sqrt{ 9x^2 + 4 y^2 + 12 \cdot(\vec{x}\cdot \vec{y}) } \\\\ && \qquad \text{unit vectors: $|\vec{x}|=x=1 \quad|\vec{y}|=y=1$} \\\\ &=& \sqrt{ 9 + 4 + 12 \cdot (\vec{x}\cdot \vec{y}) } \\\\ && \qquad \text{Formula: $\vec{x}\cdot \vec{y} = |\vec{x}|\cdot |\vec{y}| \cdot \cos(120^{\circ})=1\cdot 1\cdot (-0.5)=-0.5 $ } \\\\ &=& \sqrt{ 9 + 4 + 12 \cdot (-0.5) } \\\\ &=& \sqrt{ 13 -6 } \\\\ &\mathbf{=}& \mathbf{\sqrt{ 7 }} \\ \hline \end{array}$