Vectors in three dimensionsEdexcel A-Level Maths: Mind map

Exam questions on Vectors in three dimensions

1. Relative to an origin $O$, the points $A$ and $B$ have coordinates $(2,-1,3)$ and $(5,3,15)$.
   The point $C$ is such that $\overrightarrow{AC}=2\overrightarrow{AB}$. Find the position vector of $C$.2 marks
2. A drone flies from a point $P$ to a point $Q$. Relative to an origin $O$ at ground level, with $\mathbf{i}$ pointing east, $\mathbf{j}$ north and $\mathbf{k}$ vertically upwards (distances in metres), $P$ has position vector $4\mathbf{i}-2\mathbf{j}+7\mathbf{k}$ and $Q$ has position vector $-2\mathbf{i}+4\mathbf{j}+4\mathbf{k}$.
   Find the exact distance $PQ$.2 marks
3. Relative to an origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}=\mathbf{i}-\mathbf{j}+2\mathbf{k}$, $\mathbf{b}=3\mathbf{i}+\mathbf{j}+3\mathbf{k}$ and $\mathbf{c}=-\mathbf{i}+4\mathbf{k}$.
   Show that $AB=AC$.3 marks