Question

A boat's velocity, measured in meters per second, is described by vector b= <3,-4) In two or more complete sentences explain how to find the speed of the boat and the direction it is traveling in standard position. In your final
answer, include all of your calculations.

Answer 1

Answer:

Direction (Standard Position): 306.87° SE

Speed: 5 m/s

Step-by-step explanation:

Part 1: Direction Angle

The direction angle for a vector $v=\langle a,b\rangle$ can be found by using the formula  $\alpha=tan^{-1}(\frac{b}{a})$ and then accounting the reference angle for the quadrant the vector is located in:

$v=\langle3,-4\rangle\\\\\alpha=tan^{-1}(\frac{4}{-3})\\\\\alpha\approx-53.13^\circ$

Since $v=\langle3,-4\rangle$ is located in Quadrant IV, then the direction angle must also be located in Quadrant IV. Thus, the true direction angle is $\theta=360+\alpha$:

$\theta=360^\circ+\alpha\\\\\theta=360^\circ+(-53.13^\circ)\\\\\theta=360^\circ-53.13^\circ\\\\\theta=306.87^\circ$

This means that the direction the boat is traveling in standard position is 306.87° SE.

Part 2: Speed (Magnitude)

To determine the speed of the vector, we must determine its magnitude, which can be defined as $||v||=\sqrt{a^2+b^2}$, thus:

$||v||=\sqrt{a^2+b^2}\\\\||v||=\sqrt{(3)^2+(-4)^2}\\\\||v||=\sqrt{9+16}\\\\||v||=\sqrt{25}\\\\||v||=5$

This means that the speed of the boat is 5 m/s