5. Let ( vec{v}=langle 1,0,0 angle ) and ( vec{w}=leftlangle w_{1}, w_{2}, w_{3} ight angle ). (a) Suppose the angle ( heta ) between ( vec{v} ) and ( vec{w} ) is ( pi / 3 ). Determine an equation satisfied by the components of ( vec{w} ) (simplify this equation). (b) Determine the unit vectors ( vec{w} ) such that the angle between ( vec{v} ) and ( vec{w} ) is ( pi / 3 ).
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The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \), where \( \theta \) is the angle between the vectors and \( |\vec{a}| \) and \( |\vec{b}| \) are the magnitudes (lengths) of Show moreā¦
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Assume that $\|\mathbf{v}\|=3,$ Iw $\|=5$ and that the angle between $\mathbf{v}$ and $w$ is $\theta=\frac{\pi}{3}$ . \begin{equation}\begin{array}{l}{\text { (a) Use the relation } \| \mathbf{v}+\mathrm{wl}^{2}=(\mathbf{v}+\mathrm{w}) \cdot(\mathbf{v}+\mathrm{w}) \text { to show that }} \\ {\|\mathbf{v}+\mathrm{w}\|^{2}=3^{2}+5^{2}+2 \mathbf{v} \cdot \mathrm{w} .} \\ {\text { (b) Find }\|\mathbf{v}+\mathrm{w}\|}\end{array}\end{equation}
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