Planes and the scalar productEdexcel A-Level Further Maths: Mind map

Exam questions on Planes and the scalar product

1. Let $\mathbf p=3\mathbf i+2\mathbf j-\mathbf k$ and $\mathbf q=\mathbf i-4\mathbf j+2\mathbf k$.
   Find the angle between $\mathbf p$ and $\mathbf q$, giving your answer in degrees to 1 decimal place.2 marks
2. The plane $\Pi_1$ has vector equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}$.
   The point $(k,2,4)$ lies on $\Pi_1$. Find the value of $k$.2 marks
3. The line $l$ has equation $\mathbf r=\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}+t\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$ and the plane $\Pi$ has equation $x+2y-2z=6$.
   Find the coordinates of the point where $l$ meets $\Pi$.3 marks