Vector product and scalar triple productEdexcel A-Level Further Maths: Subtopic test

1. 1
   The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and $\mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}$.

   (a)

   Find $\mathbf{a}\times\mathbf{b}$.

   [1 mark]

   • A$-5\mathbf{i}-5\mathbf{j}+5\mathbf{k}$
   • B$2\mathbf{i}-2\mathbf{j}+3\mathbf{k}$
   • C$5\mathbf{i}-5\mathbf{j}-5\mathbf{k}$
   • D$5\mathbf{i}+5\mathbf{j}-5\mathbf{k}$

   (b)

   Find the area of the parallelogram with adjacent sides $\mathbf{a}$ and $\mathbf{b}$.

   [1 mark]

   • A$15$
   • B$5\sqrt3$
   • C$2\sqrt{21}$
   • D$75$

   (c)

   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]

   Total for question 1: 4 marks

2. 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 $\overrightarrow{OC}=2\mathbf{i}+\mathbf{k}$.

   (a)

   Find $\overrightarrow{OA}\cdot\left(\overrightarrow{OB}\times\overrightarrow{OC}\right)$.

   [1 mark]

   • A$11$
   • B$-11$
   • C$15$
   • D$-13$

   (b)

   Find the volume of the tetrahedron $OABC$.

   [1 mark]

   • A$11$
   • B$\frac{11}{3}$
   • C$\frac{11}{6}$
   • D$\frac{11}{2}$

   (c)

   Find a vector that is perpendicular to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The tetrahedron $ABCD$ has vertices $A(1,0,2)$, $B(2,3,0)$, $C(4,1,1)$ and $D(2,-1,4)$.

   (a)

   Find the exact area of triangle $ABC$.

   [3 marks]

   (b)

   Find the volume of the tetrahedron $ABCD$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A parallelepiped has one vertex at the origin $O$. The three edges from $O$ are $\overrightarrow{OP}=3\mathbf{i}+\mathbf{k}$, $\overrightarrow{OQ}=\mathbf{i}+2\mathbf{j}$ and $\overrightarrow{OR}=\mathbf{j}+4\mathbf{k}$.

   (a)

   (i) Show that the volume of the parallelepiped is $25$.
   (ii) Find the perpendicular distance from $P$ to the plane containing $O$, $Q$ and $R$.

   [6 marks]

   (b)

   (i) Find the area of triangle $PQR$.
   (ii) Use the volume of the parallelepiped, $25$, to find the volume of the tetrahedron $OPQR$, and hence find the perpendicular distance from $O$ to the plane $PQR$.

   [6 marks]

   Total for question 4: 12 marks

End of questions