Question

Form the cross product of vector A = 23km 37° and vector C = 47km 193°

Answer 1

Let's first find the $x$- and $y$-components of vectors A and C:

$$A_x=Acos\theta=23\ km\cdot cos37^{\circ}=18.37\ km,$$ $$A_y=Asin\theta=23\ km\cdot sin37^{\circ}=13.84\ km,$$ $$C_x=Ccos\theta=47\ km\cdot cos193^{\circ}=-45.8\ km,$$ $$C_y=Csin\theta=47\ km\cdot sin193^{\circ}=-10.6\ km.$$

We can find the cross product of vector A and vector C as follows:

$$\vec{A}\times \vec{C}=\begin{vmatrix} A_x & A_y \\ C_x & C_y \end{vmatrix}\cdot\vec{k},$$ $$\vec{A}\times \vec{C}=(A_xC_y-C_xA_y)\cdot\vec{k},$$ $$\vec{A}\times \vec{C}=(18.37\ km\cdot(-10.6\ km)-(-45.8\ km)\cdot13.84\ km)\cdot\vec{k},$$ $$\vec{A}\times \vec{C}=(439.15\ km^2)\cdot\vec{k}$$