Magnitude, direction and position vectorsAQA A-Level Maths: Subtopic test

1. 1
   The vector $\mathbf p=5\mathbf i-12\mathbf j$.

   (a)

   Find $|\mathbf p|$.

   [1 mark]

   • A$7$
   • B$17$
   • C$\sqrt{119}$
   • D$13$

   (b)

   The direction of $\mathbf p$ is measured anticlockwise from the positive $x$-direction. Find it, to 1 decimal place.

   [1 mark]

   • A$67.4^\circ$
   • B$112.6^\circ$
   • C$292.6^\circ$
   • D$247.4^\circ$

   (c)

   Find a unit vector in the direction of $\mathbf p$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points $A$ and $B$ have position vectors $\mathbf a=2\mathbf i+3\mathbf j$ and $\mathbf b=8\mathbf i-5\mathbf j$ relative to the origin $O$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$-6\mathbf i+8\mathbf j$
   • B$6\mathbf i-8\mathbf j$
   • C$10\mathbf i-2\mathbf j$
   • D$6\mathbf i-2\mathbf j$

   (b)

   Find the distance $AB$.

   [1 mark]

   • A$10$
   • B$14$
   • C$100$
   • D$2\sqrt7$

   (c)

   The point $C$ has position vector $3\mathbf a$. Find the exact distance $AC$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The vector $\mathbf v$ has magnitude $8$ and direction $150^\circ$, measured anticlockwise from the positive $x$-direction. The vector $\mathbf w=4\sqrt3\,\mathbf i+3\mathbf j$.

   (a)

   Express $\mathbf v$ in the form $x\mathbf i+y\mathbf j$, giving exact values of $x$ and $y$.

   [3 marks]

   (b)

   Find the magnitude and the direction of $\mathbf v+\mathbf w$, the direction being measured anticlockwise from the positive $x$-direction.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The points $A$, $B$ and $C$ have position vectors $\mathbf a=\mathbf i+\mathbf j$, $\mathbf b=5\mathbf i+3\mathbf j$ and $\mathbf c=3\mathbf i+7\mathbf j$ relative to the origin $O$.

   (a)

   Show that triangle $ABC$ is right-angled and isosceles, and find its area.

   [6 marks]

   (b)

   Given that $ABCD$ is a square, find the position vector of $D$. Hence find the exact distance $OD$ and the direction of $\overrightarrow{OD}$ measured anticlockwise from the positive $x$-direction, to 1 decimal place.

   [6 marks]

   Total for question 4: 12 marks

End of questions