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Find the dot product of the vectors shown in the diagram below

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The lengths of the vectors are $$\|\mathbf{u}\|=1 \text { and }\|\mathbf{v}\|=\sqrt{8}=2 \sqrt{2}$$ and the cosine of the angle $\theta$ between them is $$\cos \left(45^{\circ}\right)=1 / \sqrt{2}$$ Thus, it follows from Formula (12) that $$\mathbf{u} \cdot \mathbf{v}=\|\mathbf{u}\|\|\mathbf{v}\| \cos \theta=(1)(2 \sqrt{2})(1 / \sqrt{2})=2$$
by Platinum (164,290 points)

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