Vectors Question 3

The vector $\textbf{i}$ represents 1 mile per hour east, and the vector $\textbf{j}$ represents 1 mile per hour north. According to her GPS, at a particular instant, Tia is biking $30^\circ$ west of north at 16 miles per hour. One of the following vectors represents Tia's velocity, in miles per hour, at that instant. Which one?

-8\textbf{i} -8\sqrt{3}\textbf{j}

-8\textbf{i} +8\sqrt{3}\textbf{j}

8\textbf{i} +8\sqrt{3}\textbf{j}

8\sqrt{3}\textbf{i}-8\textbf{j}

8\sqrt{3}\textbf{i}+8\textbf{j}

Approach

Tia's velocity can be split into horizontal and vertical components.

Horizontal Component

\sin{30^\circ} = \frac{x}{16}

x = 16\sin{30^\circ} = 8

Vertical Component

\cos{30^\circ} = \frac{y}{16}

y = 16\sin{30^\circ} = 8\sqrt{3}

Since Tia is traveling west of north, our horizontal component is negative and our vertical component is positive.

\boxed{-8\textbf{i}+8\sqrt{3}\textbf{j}}