Vectors in three dimensionsEdexcel A-Level Maths: Subtopic test

1. 1
   Relative to an origin $O$, the points $A$ and $B$ have coordinates $(2,-1,3)$ and $(5,3,15)$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$3\mathbf{i}+4\mathbf{j}+12\mathbf{k}$
   • B$7\mathbf{i}+2\mathbf{j}+18\mathbf{k}$
   • C$-3\mathbf{i}-4\mathbf{j}-12\mathbf{k}$
   • D$3\mathbf{i}+4\mathbf{j}-12\mathbf{k}$

   (b)

   Find the distance $AB$.

   [1 mark]

   • A$19$
   • B$169$
   • C$13$
   • D$5$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A drone flies from a point $P$ to a point $Q$. Relative to an origin $O$ at ground level, with $\mathbf{i}$ pointing east, $\mathbf{j}$ north and $\mathbf{k}$ vertically upwards (distances in metres), $P$ has position vector $4\mathbf{i}-2\mathbf{j}+7\mathbf{k}$ and $Q$ has position vector $-2\mathbf{i}+4\mathbf{j}+4\mathbf{k}$.

   (a)

   Find $\overrightarrow{PQ}$.

   [1 mark]

   • A$6\mathbf{i}-6\mathbf{j}+3\mathbf{k}$
   • B$-6\mathbf{i}+6\mathbf{j}-3\mathbf{k}$
   • C$2\mathbf{i}+2\mathbf{j}+11\mathbf{k}$
   • D$-6\mathbf{i}+6\mathbf{j}+3\mathbf{k}$

   (b)

   Find the position vector of the midpoint of $PQ$.

   [1 mark]

   • A$2\mathbf{i}+2\mathbf{j}+11\mathbf{k}$
   • B$\mathbf{i}+\mathbf{j}+11\mathbf{k}$
   • C$-3\mathbf{i}+3\mathbf{j}-1.5\mathbf{k}$
   • D$\mathbf{i}+\mathbf{j}+5.5\mathbf{k}$

   (c)

   Find the exact distance $PQ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Relative to an origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}=\mathbf{i}-\mathbf{j}+2\mathbf{k}$, $\mathbf{b}=3\mathbf{i}+\mathbf{j}+3\mathbf{k}$ and $\mathbf{c}=-\mathbf{i}+4\mathbf{k}$.

   (a)

   Show that $AB=AC$.

   [3 marks]

   (b)

   Show that angle $BAC=90^\circ$ and hence find the area of triangle $ABC$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A room is a cuboid $12$ m long, $4$ m wide and $3$ m high. The origin $O$ is at a floor corner, with $\mathbf{i}$ along the length, $\mathbf{j}$ along the width and $\mathbf{k}$ vertically upwards (distances in metres). The point $G$ is the ceiling corner diagonally opposite $O$. A lamp $L$ hangs on the ceiling at position vector $4\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and a camera $C$ stands on the floor at position vector $10\mathbf{i}+3\mathbf{j}$.

   (a)

   (i) Write down the position vector of $G$ and find the distance $OG$.
   (ii) Find $\overrightarrow{LC}$ and the exact distance $LC$.
   (iii) Find the position vector of the midpoint of $LC$.

   [6 marks]

   (b)

   A bird flies in a straight line from $O$ to $G$.
   (i) The point $H$ is the point on $OG$ that is halfway between $O$ and $G$. Find the position vector of $H$.
   (ii) Find the distance $HL$.
   (iii) The bird must keep at least $3$ m from the lamp. Does the bird keep to this rule? Justify your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions