The scalar productEdexcel International A Level Maths: Subtopic test

1. 1
   The vectors $\mathbf{p}=2\mathbf{i}-3\mathbf{j}+\mathbf{k}$ and $\mathbf{q}=4\mathbf{i}+\mathbf{j}-5\mathbf{k}$ are given.

   (a)

   Find $\mathbf{p}\cdot\mathbf{q}$.

   [1 mark]

   • A$6$
   • B$16$
   • C$0$
   • D$10$

   (b)

   What can be deduced from the value of $\mathbf{p}\cdot\mathbf{q}$?

   [1 mark]

   • A$\mathbf{p}$ and $\mathbf{q}$ are parallel
   • B$\mathbf{p}$ and $\mathbf{q}$ are perpendicular
   • C$\mathbf{p}$ and $\mathbf{q}$ have the same magnitude
   • DThe angle between $\mathbf{p}$ and $\mathbf{q}$ is $180^\circ$

   (c)

   The vector $\mathbf{r}=\mathbf{i}+2\mathbf{j}+\lambda\mathbf{k}$ is perpendicular to $\mathbf{p}$. Find the value of $\lambda$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Relative to the origin $O$, the point $A$ has position vector $2\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and the point $B$ has position vector $3\mathbf{i}-6\mathbf{j}+2\mathbf{k}$.

   (a)

   Find $|\overrightarrow{OB}|$.

   [1 mark]

   • A$\sqrt{45}$
   • B$49$
   • C$\sqrt{13}$
   • D$7$

   (b)

   Find $\overrightarrow{OA}\cdot\overrightarrow{OB}$.

   [1 mark]

   • A$-4$
   • B$16$
   • C$20$
   • D$-8$

   (c)

   Find the size of angle $AOB$, giving your answer in degrees to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The points $A$, $B$ and $C$ have coordinates $A(1,-1,2)$, $B(3,1,3)$ and $C(2,-3,0)$.

   (a)

   Find the size of angle $BAC$, giving your answer in degrees to 1 decimal place.

   [3 marks]

   (b)

   Find the size of angle $ABC$, giving your answer in degrees to 1 decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The points $A$, $B$ and $C$ have coordinates $A(1,2,3)$, $B(4,-1,5)$ and $C(3,4,p)$, where $p$ is a constant. Angle $BAC$ is $90^\circ$.

   (a)

   Find the value of $p$, and hence find the exact area of triangle $ABC$.

   [6 marks]

   (b)

   The point $D$ is such that $ABDC$ is a rectangle. Using the value of $p$ from (a), find the coordinates of $D$, and find the acute angle between the diagonals $AD$ and $BC$, giving your answer in degrees to 1 decimal place.

   [6 marks]

   Total for question 4: 12 marks

End of questions