IB Maths AA HL 3.16 Vector product

[Maximum number: 5]

The following question compares the distance and direction between cities on a flat surface to the distance and direction between cities on a sphere.
Consider a model where the cities of Bogotá, Moscow, and Nairobi lie on a flat surface. In this model, Nairobi is 6000 km due south of Moscow and Bogotá is 12500 km due west of Nairobi, as shown in the following diagram.

Figure for Question IB Maths AA HL Ahl 3 16 Hl Vector Product Questions question 1 — IB Maths AA HL

(a)

Find the vector $\boldsymbol{a} \times \boldsymbol{p}$ .

[ 2 ]

attempt to find cross product (e.g. correct except for sign error)
$\boldsymbol{a} \times \boldsymbol{p}=\left(\begin{array}{c}0 \\ -36 \\ 0\end{array}\right)$

(b)

Show that the angle at vertex A in the spherical triangle is $90^{\circ}$ .

Moscow, M , has position vector $\overrightarrow{\mathrm{OM}}=\boldsymbol{m}=\left(\begin{array}{c}0 \\ 6 \cos \theta \\ 6 \sin \theta\end{array}\right)$ , as shown in the following diagram.

Figure for Question (b) — IB Maths AA HL

The shortest distance between two points on the sphere lies along an arc of a circle on the sphere with centre O . In this model the shortest distance from Moscow to Nairobi is 6000 km .

[ 3 ]

METHOD 1

\boldsymbol{a} \times \boldsymbol{n}=\left(\begin{array}{c} 0 \\ 0 \\ 36 \end{array}\right)

choosing to use scalar product for their $\boldsymbol{a} \times \boldsymbol{p}$ and $\boldsymbol{a} \times \boldsymbol{n}$ $(\boldsymbol{a} \times \boldsymbol{p}) \cdot(\boldsymbol{a} \times \boldsymbol{n})=\mathbf{0}$
so angle is $90^{\circ}$

METHOD 2
$\boldsymbol{a} \times \boldsymbol{n}=\left(\begin{array}{c}0 \\ 0 \\ 36\end{array}\right)$

Note: The final A1 can be awarded for seeing the dot product equals zero without incorrect working. Follow through from (i) is permitted and if they explicitly state that they are using $\boldsymbol{a} \times \boldsymbol{n}$ then allow internal follow through. \\ Note: The final A1 can be awarded for seeing the dot product equals zero
without incorrect working. Follow through from (i) is permitted and if they
explicitly state that they are using $\boldsymbol{a} \times \boldsymbol{n}$ then allow internal follow through.
o angle is $90^{\circ}$
attempt to use cross product for their $\boldsymbol{a} \times \boldsymbol{p}$ and $\boldsymbol{a} \times \boldsymbol{n}$ . This must include both an attempt to find the cross product and knowledge that $\sin \theta=\frac{|x \times y|}{|x||y|}$ $(\boldsymbol{a} \times \boldsymbol{n}) \times(\boldsymbol{a} \times \boldsymbol{p})=\left(\begin{array}{c}1296 \\ 0 \\ 0\end{array}\right)$ OR $(\boldsymbol{a} \times \boldsymbol{p}) \times(\boldsymbol{a} \times \boldsymbol{n})=\left(\begin{array}{c}-1296 \\ 0 \\ 0\end{array}\right)$ $\sin \theta=\frac{1296}{36 \times 36}=1$
so angle is $90^{\circ}$
AG

IB Maths AA HL Ahl 3 16 Hl Vector Product Questions question 1

Question (a)

Question (b)