The vector productEdexcel International A Level Further Maths: Subtopic test

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

   (a)

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

   [1 mark]

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

   (b)

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

   [1 mark]

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

   (c)

   Find the area of the triangle with sides $\mathbf{a}$ and $\mathbf{b}$, and write down a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ with magnitude $1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A parallelepiped has edges represented by the vectors $\mathbf{a}=\mathbf{i}+\mathbf{j}+2\mathbf{k}$, $\mathbf{b}=3\mathbf{i}+\mathbf{j}$ and $\mathbf{c}=2\mathbf{j}+\mathbf{k}$.

   (a)

   Find the value of $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$.

   [1 mark]

   • A$16$
   • B$-10$
   • C$10$
   • D$4$

   (b)

   A tetrahedron has three edges represented by $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$. Find its volume.

   [1 mark]

   • A$10$
   • B$5$
   • C$\frac{10}{3}$
   • D$\frac53$

   (c)

   The vector $\mathbf{d}=2\mathbf{j}+\lambda\mathbf{k}$ is such that $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{d}$ lie in the same plane. Find the value of $\lambda$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The points $P$, $Q$, $R$ and $S$ have coordinates $(1,2,0)$, $(3,3,2)$, $(2,5,1)$ and $(0,1,4)$ respectively.

   (a)

   Find the exact area of triangle $PQR$.

   [3 marks]

   (b)

   Find the exact volume of the tetrahedron $PQRS$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Relative to an origin $O$, the points $A$, $B$, $C$ and $D$ have coordinates $(1,0,2)$, $(2,2,4)$, $(0,0,4)$ and $(5,-1,4)$ respectively.

   (a)

   Find $\overrightarrow{AB}\times\overrightarrow{AC}$. Hence find the area of triangle $ABC$ and the shortest distance from $C$ to the line $AB$.

   [6 marks]

   (b)

   Find the volume of the tetrahedron $ABCD$, and hence the shortest distance from $D$ to the plane $ABC$.

   [6 marks]

   Total for question 4: 12 marks

End of questions