Vector product and scalar triple productEdexcel A-Level Further Maths: Mind map

Exam questions on Vector product and scalar triple product

1. The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and $\mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}$.
   The points $O$, $A$ and $B$ have position vectors $\mathbf{0}$, $\mathbf{a}$ and $\mathbf{b}$. Find the exact area of triangle $OAB$.2 marks
2. The tetrahedron $OABC$ has $O$ at the origin, and $\overrightarrow{OA}=\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\overrightarrow{OB}=\mathbf{j}+3\mathbf{k}$ and $\overrightarrow{OC}=2\mathbf{i}+\mathbf{k}$.
   Find a vector that is perpendicular to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$.2 marks
3. The tetrahedron $ABCD$ has vertices $A(1,0,2)$, $B(2,3,0)$, $C(4,1,1)$ and $D(2,-1,4)$.
   Find the exact area of triangle $ABC$.3 marks