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Question

# Find the Euclidean distance between u a nd v and the cosine of the angle between those vectors. State whether that angle is acute, obtuse, or 90°. (a) u = (3, 3, 3), v = (1, 0, 4)

Solution

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The Euclidean distance between $\textbf{u}$ and $\textbf{v}$ given by

\begin{align*} \|\textbf{u}-\textbf{v}\|=\sqrt{(3-1)^2+(3-0)^2+(3-4)^2}=\sqrt{4+9+1}=\sqrt{14} \end{align*}

Let $\theta$ be the angle between $\textbf{u}$ and $\textbf{v}$. Then the cosine of the angle between $\textbf{u}$ and $\textbf{v}$ is given by:

\begin{align*} \cos{\theta} &=\frac{\textbf{u}\cdot\textbf{v}}{\|\textbf{u}\|\|\textbf{v}\|}=\frac{(3,3,3)\cdot (1,0,4)}{\|(3,3,3)\|\|(1,0,4)\|}\\ &=\frac{3+0+12}{\sqrt{3^2+3^2+3^2}\sqrt{1^2+0+4^2}}\\ &=\frac{15}{\sqrt{27}\sqrt{17}} \end{align*}

Then $\theta=\cos^{-1}\left(\frac{15}{\sqrt{27}\sqrt{17}}\right) \approx 45.56^{\circ}<90^{\circ}$. Thus $\theta$ is an acute angle.

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