Vectors in three dimensionsAQA A-Level Maths: Subtopic test

1. 1
   The vectors $\mathbf a=2\mathbf i-3\mathbf j+6\mathbf k$ and $\mathbf b=\mathbf i+2\mathbf j-2\mathbf k$.

   (a)

   Find $|\mathbf a|$.

   [1 mark]

   • A$11$
   • B$49$
   • C$7$
   • D$\sqrt{13}$

   (b)

   Find $2\mathbf a-3\mathbf b$.

   [1 mark]

   • A$7\mathbf i+6\mathbf k$
   • B$\mathbf i-12\mathbf j+18\mathbf k$
   • C$\mathbf i-12\mathbf j+6\mathbf k$
   • D$4\mathbf i-13\mathbf j+22\mathbf k$

   (c)

   Find the exact distance between the points with position vectors $\mathbf a$ and $\mathbf b$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points $A$ and $B$ have position vectors $\mathbf a=\mathbf i+2\mathbf j+3\mathbf k$ and $\mathbf b=4\mathbf i-2\mathbf j+15\mathbf k$ relative to the origin $O$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$-3\mathbf i+4\mathbf j-12\mathbf k$
   • B$5\mathbf i+18\mathbf k$
   • C$3\mathbf i-4\mathbf j+18\mathbf k$
   • D$3\mathbf i-4\mathbf j+12\mathbf k$

   (b)

   Find the distance $AB$.

   [1 mark]

   • A$13$
   • B$19$
   • C$169$
   • D$\sqrt{119}$

   (c)

   The point $C$ is such that $\overrightarrow{AC}=2\overrightarrow{AB}$. Find the position vector of $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $\mathbf u=2\mathbf i-\mathbf j+4\mathbf k$, $\mathbf v=-6\mathbf i+3\mathbf j+\mu\mathbf k$ and $\mathbf w=2\mathbf i+3\mathbf j+q\mathbf k$, where $\mu$ and $q$ are constants.

   (a)

   Given that $\mathbf u$ and $\mathbf v$ are parallel, find the value of $\mu$.

   [3 marks]

   (b)

   Given that $|\mathbf u+\mathbf w|=6$, find the possible values of $q$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Relative to a fixed origin $O$, the unit vectors $\mathbf i$, $\mathbf j$ and $\mathbf k$ point east, north and vertically upwards, and distances are in metres. At time $t=0$ seconds, drone $A$ is at the point with position vector $(5\mathbf i-4\mathbf j+10\mathbf k)$ m and moves with constant velocity $(2\mathbf i+3\mathbf j+6\mathbf k)$ m s⁻¹. At the same time drone $B$ is at $(17\mathbf i+2\mathbf j+16\mathbf k)$ m and moves with constant velocity $(-2\mathbf i+\mathbf j+4\mathbf k)$ m s⁻¹.

   (a)

   (i) Find the speed of drone $A$. (ii) Find the position vector of $A$ when $t=4$. (iii) Find the time at which $A$ is $70$ m above $O$.

   [6 marks]

   (b)

   Show that the two drones collide, and find the position vector of the point of collision.

   [6 marks]

   Total for question 4: 12 marks

End of questions