The Bonneville Salt Flats, located in Utah near the border with Nevada, not far from interstate I80, cover an area of over 30000 acres. A race car driver on the Flats first heads north for 6.71 km, then makes a sharp turn and heads southwest for 1.33 km, then makes another turn and heads east for 3.67 km. How far is she from where she started?

We can put the starting point at the origin, let the x-axis point to the East and the y-axis North. We can associate each displacement with a vector, considering that the second vector starts where the first ended and the third vector started where the second ended, it is pretty straightforward to know that the finishing location with respect to the starting place is given by the sum of the three vectors, thus we can determine the modulus of the sum vector to know the distance between the destination and the starting point. See the diagram for details, here is the step by step solution:

First we define the three displacement vectors vectors

[tex]\vec{A}(0;6.71)[/tex]

[tex]\vec{B}(-1.33\sin{45};-1.33\cos{45})[/tex]

[tex]\vec{C}(3.67;0)[/tex]

Now adding the we get:

[tex]\vec{A}+\vec{B}+\vec{C}=(3.67 - 1.33\sin{45}; 6.71 -1.33\cos{45})[/tex]

We then calculate the modulus of this vector:

[tex]|\vec{A}+\vec{B}+\vec{C}|=\sqrt{(3.67-1.33\sin{45})^2+(6.71-1.33\cos{45})^2}\approx 6.38\, Km[/tex]