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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)

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Answered 1 year ago
Answered 1 year ago
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The Euclidean distance between u\textbf{u} and v\textbf{v} given by

uv=(31)2+(30)2+(34)2=4+9+1=14\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 θ\theta be the angle between u\textbf{u} and v\textbf{v}. Then the cosine of the angle between u\textbf{u} and v\textbf{v} is given by:

cosθ=uvuv=(3,3,3)(1,0,4)(3,3,3)(1,0,4)=3+0+1232+32+3212+0+42=152717\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 θ=cos1(152717)45.56<90\theta=\cos^{-1}\left(\frac{15}{\sqrt{27}\sqrt{17}}\right) \approx 45.56^{\circ}<90^{\circ}. Thus θ\theta is an acute angle.

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