3.13 Scalar and vector productsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Two forces $\mathbf{a}=\begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}$ N and $\mathbf{b}=\begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}$ N act on a particle.

   (a)

   Calculate $\mathbf{a}\cdot\mathbf{b}$.

   [1 mark]

   • A$\begin{pmatrix} 3 \\ -2 \\ 8 \end{pmatrix}$
   • B$9$
   • C$13$
   • D$\sqrt{294}$

   (b)

   Find the angle between $\mathbf{a}$ and $\mathbf{b}$.

   [1 mark]

   • A$31.7^\circ$
   • B$121.7^\circ$
   • C$0.525^\circ$
   • D$58.3^\circ$

   (c)

   A third force $\mathbf{c}=\begin{pmatrix} k \\ -1 \\ 2 \end{pmatrix}$ N is perpendicular to $\mathbf{a}$. Find the value of $k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A triangular roof panel has corners $A(1,0,2)$, $B(4,2,1)$ and $C(2,3,5)$, where the coordinates are in metres.

   (a)

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

   [1 mark]

   • A$\begin{pmatrix} -9 \\ 10 \\ -7 \end{pmatrix}$
   • B$\begin{pmatrix} 9 \\ 10 \\ 7 \end{pmatrix}$
   • C$\begin{pmatrix} 9 \\ -10 \\ 7 \end{pmatrix}$
   • D$\begin{pmatrix} 3 \\ 6 \\ -3 \end{pmatrix}$

   (b)

   Find the area of the panel.

   [1 mark]

   • A$7.58$ m$^2$
   • B$15.2$ m$^2$
   • C$115$ m$^2$
   • D$3.74$ m$^2$

   (c)

   Find the shortest distance from $C$ to the line through $A$ and $B$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two straight pipes are modelled by the lines $l_1:\ \mathbf{r}=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+s\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$ and $l_2:\ \mathbf{r}=\begin{pmatrix} 3 \\ -1 \\ 0 \end{pmatrix}+t\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$, where the units are metres.

   (a)

   Use your GDC to find the acute angle between the two pipes.

   [3 marks]

   (b)

   A force $\mathbf{F}=\begin{pmatrix} 4 \\ -2 \\ 3 \end{pmatrix}$ N acts at a point on $l_2$. Find the component of $\mathbf{F}$ (i) along the direction of $l_2$, (ii) perpendicular to $l_2$, in the plane of $\mathbf{F}$ and $l_2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A solar panel is modelled as the parallelogram $OABC$, where $O$ is the origin and, in metres, $\overrightarrow{OA}=\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\overrightarrow{OC}=\begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$.

   (a)

   (i) Calculate $\overrightarrow{OA}\cdot\overrightarrow{OC}$.
   (ii) Hence find the angle $A\hat{O}C$.
   (iii) Find $\overrightarrow{OA}\times\overrightarrow{OC}$ and hence the area of the panel.

   [6 marks]

   (b)

   (i) Find a unit vector perpendicular to the panel.
   (ii) Find the shortest distance from $A$ to the line $OC$.
   (iii) The wind exerts a force $\mathbf{F}=\begin{pmatrix} 0 \\ 0 \\ 12 \end{pmatrix}$ N. Find the component of $\mathbf{F}$ acting perpendicular to the panel.

   [6 marks]

   Total for question 4: 12 marks

End of questions