Use the scalar (dot) product. 𝒂 ⋅ 𝒃 = |𝒂||𝒃| cos(𝜃) ( 𝑎1 𝑎2 𝑎3) ⋅ ( 𝑏1 𝑏2 𝑏3) = 𝑎1 𝑏1 + 𝑎2 𝑏2 + 𝑎3 𝑏3

Calculate the angle between the vectors \(\mathbf{u} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}\).

Two forces acting on a particle are represented by vectors α = (2, −1, 3) and β = (−3, 4, 1). a) Calculate the scalar product α ⋅ β using the component form. (1) b) Hence, use the scalar product to determine the angle between the two force vectors, to the nearest degree. (1)

Two forces act on a particle at the origin. Force $\vec{F}_1$ has magnitude $10$ N and acts in the direction of vector $(3, 0, 4)$. Force $\vec{F}_2$ has magnitude $5$ N and acts in the direction of vector $(0, 1, 0)$. Calculate the angle between the two force vectors, correct to the nearest degree.

Consider the vectors \(\mathbf{p} = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} 4 \\ 1 \\ -3 \end{pmatrix}\). a) Determine the value of \(\mathbf{p} \cdot \mathbf{q}\). (1 mark) b) Use your result from part a) to determine the angle \(\theta\) between vectors \(\mathbf{p}\) and \(\mathbf{q}\). Give your answer in degrees, correct to 2 decimal places. (2 marks)