Angle between vectors

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Which formula is used to find the angle $\theta$ between two nonzero vectors $\mathbf{a}$ and $\mathbf{b}$ ?

\theta=\cos^{-1}\left(\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right)

\theta=\cos^{-1}\left(\dfrac{|\mathbf{a}\times\mathbf{b}|}{|\mathbf{a}||\mathbf{b}|}\right)

\theta=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}

(no inverse cosine needed)

\theta=\sin^{-1}\left(\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right)

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If $\overrightarrow{OA}=2\mathbf{i}+3\mathbf{j}-\mathbf{k}$ and $\overrightarrow{OB}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}$ , determine the cosine of the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ .

\cos\theta=-0.5

(so

\theta=120^\circ

)

\cos\theta=0.5

(so

\theta=60^\circ

)

\cos\theta=-7

\cos\theta=-0.5

(so

\theta=60^\circ

)

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A force $\mathbf{F}=3\mathbf{i}-4\mathbf{j}$ N acts on an object as it is displaced through $\mathbf{d}=-2\mathbf{i}-\mathbf{j}+2\mathbf{k}$ m. Find the angle between $\mathbf{F}$ and $\mathbf{d}$ .

\theta\approx97.7^\circ

\theta\approx82.3^\circ

\theta\approx7.7^\circ

\theta\approx90^\circ

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Angle between vectors

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