IB Maths AA HL 3.13 Scalar product Question Bank

METHOD 1

EITHER (Scalar product)
attempt to use scalar product to find BÔM $=\arccos \left(\frac{\boldsymbol{b} . \boldsymbol{m}}{|\boldsymbol{b}||\boldsymbol{m}|}\right)$

Note: This may be written as $\cos ($ BÔM $)=\frac{\boldsymbol{b} . \boldsymbol{m}}{|\boldsymbol{b}||\boldsymbol{m}|}$.
$=\arccos \left(\frac{\left(6 \cos 120^{\circ}\right)\left(6 \cos 57.3^{\circ}\right)}{6^{2}}\right)$

Note: This (A1) can be awarded for $\cos \theta=\frac{6 \cos 120^{\circ} \times 6 \cos 57.3^{\circ}}{6^{2}}$.
$=106^{\circ}\left(=105.673 \ldots{ }^{\circ}\right)$ OR 1.84 radians $(1.84434 \ldots)$

OR (Cosine Rule)
$|\mathrm{BM}|=\sqrt{(6 \sin 120-0)^{2}+(6 \cos 120-6 \sin 57.3)^{2}+(0-6 \sin 57.3)^{2}}$$=9.56(9.56299 \ldots)$ OR 9560 km
attempt to use the cosine rule to find angle BOM
$|\mathrm{BM}|^{2}=6^{2}+6^{2}-2 \times 6 \times 6 \times \cos \mathrm{BOM}$$\mathrm{BOM}=106^{\circ}(=105.673 \ldots)$ OR 1.84 radians $(1.84434 \ldots)$

THEN
attempt to use $\frac{\theta}{360} \times 2 \pi r$ OR $s=r \theta$$=\frac{105.671^{\circ}}{360} \times 2 \pi \times 6$$=11.1(11.0660 \ldots$ thousand km)

Note: Accept an answer of 11100 OR $11066.0 \ldots$ without units (i.e. "km" is implicit).

METHOD 2 (Cross product)
attempt to use cross product to find BÔM $=\arcsin \frac{|\boldsymbol{b} \times \boldsymbol{m}|}{|\boldsymbol{b}||\boldsymbol{m}|}$

Note: This may be written as $\sin$ BÔM $=\frac{|\boldsymbol{b} \times \boldsymbol{m}|}{|\boldsymbol{b}||\boldsymbol{m}|}$.

Note: Award at most (M1)(A1)(A0)(M1)A0 for an answer of 7.78 thousand km from finding the principal root of the arcsin.

attempt to calculate $\boldsymbol{b} \times \boldsymbol{m}$$\boldsymbol{b} \times \boldsymbol{m}=\left(\begin{array}{c}36 \cos 120^{\circ} \sin 57.3^{\circ} \\ -36 \sin 120^{\circ} \sin 57.3^{\circ} \\ 36 \sin 120^{\circ} \cos 57.3^{\circ}\end{array}\right)\left(=\left(\begin{array}{c}-15.1464 \ldots \\ -26.2344 \ldots \\ 16.8449 \ldots\end{array}\right)\right)$
attempt to use scalar product to find angle between vectors

Note: Award (A1) for numerator, (A1) for denominator.

Note: Do not penalise absence of degree symbol.
Award full marks for answer in radians =0.507.
Award full marks for $029^{\circ}$.
Special Case: If a candidate calculates $\boldsymbol{m} \times \boldsymbol{b}$ instead of $\boldsymbol{b} \times \boldsymbol{m}$ then you can award at most (M1)(A0)(M1)(A1)(A1)A0.