Consider a map oriented so that the x axis runs east–west (with east being the "positive" direction) and y runs north–south (with north "positive"). A person drives 14 km to the west, then turns and drives 44 km to the south. Find the magnitude and the direction of the total movement of the driver. Express the direction as an angle measured counterclockwise from the positive x axis.

Find:

magnitude_______ km
direction______ °

To find the magnitude and direction of the total movement of the driver, we can use the Pythagorean theorem and trigonometric functions.

1) Magnitude:
The magnitude represents the total distance traveled by the driver. We can determine this by calculating the hypotenuse of a right triangle formed by the two legs of the journey (west and south).

Using the Pythagorean theorem, we have:
magnitude = √(west^2 + south^2)

Given that the driver drove 14 km to the west (negative x-axis direction) and then 44 km to the south (negative y-axis direction), we have:
west = -14 km
south = -44 km

Substituting the values into the formula:
magnitude = √((-14)^2 + (-44)^2)
= √(196 + 1936)
= √(2132)
≈ 46.17 km

So, the magnitude of the total movement of the driver is approximately 46.17 km.

2) Direction:
To find the direction, we need to calculate the angle measured counterclockwise from the positive x-axis. We can use the inverse tangent function (tan^(-1)) to find the angle between the resultant vector and the positive x-axis.

Direction = tan^(-1)(south/west)

Using the given values:
Direction = tan^(-1)(-44/-14)
= tan^(-1)(44/14)
≈ 71.56°

So, the direction of the total movement of the driver is approximately 71.56° counterclockwise from the positive x-axis.