FP3: VectorsEdexcel International A Level Further Maths: Topic test

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

   (a)

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

   [1 mark]

   • A$3\mathbf{i}+5\mathbf{j}+7\mathbf{k}$
   • B$-3\mathbf{i}+5\mathbf{j}-7\mathbf{k}$
   • C$-3\mathbf{i}-5\mathbf{j}-7\mathbf{k}$
   • D$3\mathbf{i}-2\mathbf{j}-2\mathbf{k}$

   (b)

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

   [1 mark]

   • A$0$
   • B$\sqrt{83}$
   • C$14$
   • D$-1$

   (c)

   Find the exact area of the triangle with sides represented by $\mathbf{a}$ and $\mathbf{b}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The vectors $\mathbf{p}=2\mathbf{i}-\mathbf{j}+\mathbf{k}$, $\mathbf{q}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}$ and $\mathbf{r}=\mathbf{j}+4\mathbf{k}$ represent three edges of a parallelepiped that meet at a vertex.

   (a)

   Find the value of $\mathbf{p}\cdot(\mathbf{q}\times\mathbf{r})$.

   [1 mark]

   • A$25$
   • B$-33$
   • C$14$
   • D$33$

   (b)

   Find the volume of the tetrahedron whose edges from the same vertex are represented by $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$.

   [1 mark]

   • A$33$
   • B$\frac{11}{2}$
   • C$11$
   • D$\frac{33}{2}$

   (c)

   The vector $\mathbf{r}$ is replaced by $\mathbf{r}+2\mathbf{p}$. State the new volume of the parallelepiped and give a reason for your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l$ has equation $\mathbf{r}=(\mathbf{i}-2\mathbf{j}+2\mathbf{k})+\lambda(2\mathbf{i}+\mathbf{j}+3\mathbf{k})$ and the plane $\Pi$ has equation $x+2y-2z=-9$.

   (a)

   Find the coordinates of the point where $l$ meets $\Pi$.

   [3 marks]

   (b)

   The point $A$ has coordinates $(1,0,4)$. Find the perpendicular distance from $A$ to $\Pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The lines $l_1$ and $l_2$ have equations $\mathbf{r}=2\mathbf{i}+\mathbf{j}+s(\mathbf{i}+2\mathbf{j}-\mathbf{k})$ and $\mathbf{r}=5\mathbf{i}-2\mathbf{j}+\mathbf{k}+t(2\mathbf{i}+\mathbf{k})$ respectively.

   (a)

   Find the shortest distance between $l_1$ and $l_2$, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   The plane $\Pi$ contains $l_1$ and is parallel to $l_2$. Find an equation of $\Pi$ in the form $\mathbf{r}\cdot\mathbf{n}=p$. Hence, by finding the distance from a point on $l_2$ to $\Pi$, confirm your answer to part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The points $A(2,1,0)$, $B(4,2,-1)$ and $C(3,4,2)$ are given.

   (a)

   Find $\overrightarrow{AB}\times\overrightarrow{AC}$.

   [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 exact area of triangle $ABC$.

   [1 mark]

   • A$5\sqrt3$
   • B$\frac{15}{2}$
   • C$\frac52$
   • D$\frac{5\sqrt3}{2}$

   (c)

   Find a Cartesian equation of the plane through $A$, $B$ and $C$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The plane $\Pi$ has vector equation $\mathbf{r}=\mathbf{i}+\mathbf{j}+s(\mathbf{i}-\mathbf{j}+2\mathbf{k})+t(2\mathbf{i}+\mathbf{k})$.

   (a)

   Find a vector normal to $\Pi$.

   [1 mark]

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

   (b)

   Find a Cartesian equation of $\Pi$.

   [1 mark]

   • A$-x+3y+2z=0$
   • B$-x+3y+2z=-2$
   • C$-x+3y+2z=2$
   • D$x+3y+2z=4$

   (c)

   Find the perpendicular distance from the origin to $\Pi$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The line $l$ has equation $\left(\mathbf{r}-(2\mathbf{i}+\mathbf{j}-\mathbf{k})\right)\times(\mathbf{i}-2\mathbf{j}+2\mathbf{k})=\mathbf{0}$.

   (a)

   Write the equation of $l$ in the form $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$, and find its Cartesian equations.

   [3 marks]

   (b)

   The point $P$ has coordinates $(4,0,1)$. Find the perpendicular distance from $P$ to $l$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The planes $\Pi_1$ and $\Pi_2$ have equations $2x-y+z=5$ and $x+y-2z=1$ respectively.

   (a)

   Find a vector equation of the line of intersection of $\Pi_1$ and $\Pi_2$, and give its Cartesian equations.

   [6 marks]

   (b)

   The point $P$ has coordinates $(1,1,1)$. Find an equation of the plane $\Pi_3$ that contains the line of intersection of $\Pi_1$ and $\Pi_2$ and passes through $P$, giving your answer in the form $ax+by+cz=d$.

   [6 marks]

   Total for question 8: 12 marks

End of questions