Vector productAQA A-Level Further Maths: Subtopic test

1. 1
   The vectors $\mathbf a=2\mathbf i-\mathbf j+3\mathbf k$ and $\mathbf b=\mathbf i+4\mathbf j-2\mathbf k$ are given.

   (a)

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

   [1 mark]

   • A$-10\mathbf i+7\mathbf j+9\mathbf k$
   • B$10\mathbf i-7\mathbf j-9\mathbf k$
   • C$-10\mathbf i-7\mathbf j+9\mathbf k$
   • D$-8$

   (b)

   Find the area of the parallelogram with adjacent sides $\mathbf a$ and $\mathbf b$.

   [1 mark]

   • A$\sqrt{294}$
   • B$\frac{\sqrt{230}}{2}$
   • C$230$
   • D$\sqrt{230}$

   (c)

   Show that $\mathbf a\times\mathbf b$ is perpendicular to $\mathbf b$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Triangle $ABC$ has vertices $A(1,0,2)$, $B(3,2,3)$ and $C(2,-1,5)$.

   (a)

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

   [1 mark]

   • A$\begin{pmatrix}-7 \\ 5 \\ 4\end{pmatrix}$
   • B$\begin{pmatrix}7 \\ -5 \\ -4\end{pmatrix}$
   • C$\begin{pmatrix}7 \\ 5 \\ -4\end{pmatrix}$
   • D$\begin{pmatrix}2 \\ -2 \\ 3\end{pmatrix}$

   (b)

   Find the area of triangle $ABC$.

   [1 mark]

   • A$3\sqrt{10}$
   • B$\frac{3\sqrt{11}}{2}$
   • C$\frac{3\sqrt{10}}{2}$
   • D$45$

   (c)

   Find the exact perpendicular distance from $C$ to the line $AB$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l_1$ has equation $(\mathbf r-\mathbf a)\times\mathbf b=\mathbf 0$, where $\mathbf a=\begin{pmatrix}1 \\ 2 \\ -1\end{pmatrix}$ and $\mathbf b=\begin{pmatrix}2 \\ -1 \\ 3\end{pmatrix}$.

   (a)

   Show that the point $(5,0,5)$ lies on $l_1$.

   [3 marks]

   (b)

   The line $l_2$ passes through the point with position vector $\mathbf a$ and is perpendicular to both $l_1$ and the vector $\mathbf i$. Find an equation for $l_2$ in the form $(\mathbf r-\mathbf a)\times\mathbf d=\mathbf 0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Triangle $PQR$ has vertices $P(0,1,2)$, $Q(2,3,1)$ and $R(1,0,4)$, with coordinates in metres.

   (a)

   (i) Find $\overrightarrow{PQ}\times\overrightarrow{PR}$.
   (ii) Hence find the area of triangle $PQR$.
   (iii) Find the exact shortest distance from $R$ to the line $PQ$.

   [6 marks]

   (b)

   A fourth point $S$ is added so that $PQSR$ is a parallelogram.
   (i) Find the coordinates of $S$.
   (ii) Find the area of the parallelogram.
   (iii) Show that the area is also equal to $\frac12|\overrightarrow{PS}\times\overrightarrow{QR}|$.

   [6 marks]

   Total for question 4: 12 marks

End of questions