In three‑dimensional space vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $|\mathbf{a}|=3$, $|\mathbf{b}|=4$ and $\mathbf{a}\cdot\mathbf{b}=6$. Find (i) the magnitude of $2\mathbf{a}-\mathbf{b}$ and (ii) the angle between $2\mathbf{a}-\mathbf{b}$ and $\mathbf{a}$.

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Magnitude = 2\sqrt{7}, Angle = \cos^{-1}\left(\frac{2}{\sqrt{7}}\right)

Neco 2020 Further Mathematics Question 4

In three‑dimensional space vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}|=3\), \(|\mathbf{b}|=4\) and \(\mathbf{a}\cdot\mathbf{b}=6\). Find (i) the magnitude of \(2\mathbf{a}-\mathbf{b}\) and (ii) the angle between \(2\mathbf{a}-\mathbf{b}\) and \(\mathbf{a}\).

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