Scalar product and perpendicular vectorsAQA A-Level Further Maths: Mind map

Exam questions on Scalar product and perpendicular vectors

1. The vectors $\mathbf a=\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf b=\begin{pmatrix} 4 \\ 5 \\ 1 \end{pmatrix}$ are given.
   Find the angle between $\mathbf a$ and $\mathbf b$, giving your answer to the nearest $0.1^\circ$.2 marks
2. Triangle $PQR$ has vertices $P(1,2,3)$, $Q(4,0,5)$ and $R(3,4,2)$.
   Hence find the exact area of triangle $PQR$.2 marks
3. The lines $l_1$ and $l_2$ have equations $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$ and $\mathbf r=\begin{pmatrix} 3 \\ -1 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 1 \\ 4 \\ 8 \end{pmatrix}$.
   Find the acute angle between $l_1$ and $l_2$, to the nearest $0.1^\circ$.3 marks