Vectors $\mathbf{u} = (x,2,-1)$ and $\mathbf{v} = (1,y,3)$ satisfy $\mathbf{u}\cdot\mathbf{v}=4$ and $|\mathbf{u}| = 5$. Find all possible ordered pairs $(x,y)$ and, for each pair, determine the angle between $\mathbf{u}$ and $\mathbf{v}$ (to one decimal place).

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(x,y) = (2\sqrt{5}, (7-2\sqrt{5})/2) with angle ≈ 76.4°, or (x,y) = (-2\sqrt{5}, (7+2\sqrt{5})/2) with angle ≈ 82.9°

Gce Model Questions Vol1 2020 Mathematics Question 4

Vectors \(\mathbf{u} = (x,2,-1)\) and \(\mathbf{v} = (1,y,3)\) satisfy \(\mathbf{u}\cdot\mathbf{v}=4\) and \(|\mathbf{u}| = 5\). Find all possible ordered pairs \((x,y)\) and, for each pair, determine the angle between \(\mathbf{u}\) and \(\mathbf{v}\) (to one decimal place).

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