Final answer:
The magnitude of the vector with x and y components of -38.0 and -60.0 units respectively is 70.71 units, and its direction is approximately 237.5°, taking into account that it lies in the third quadrant.
Step-by-step explanation:
The magnitude and direction of a vector can be found by using its x and y components. In this case, the given vector has components -38.0 units in the x direction and -60.0 units in the y direction.
First, to find the magnitude of the vector, we can use the Pythagorean theorem:
- Calculate the square of each component: (-38.0)^2 and (-60.0)^2.
- Add these squared values: 1444 + 3600.
- Take the square root of the sum: √(1444 + 3600) = 70.71 units.
Therefore, the magnitude of the vector is 70.71 units.
To determine the direction of the vector, we find the angle it makes with the positive x-axis using the tangent function and arctan (also known as tan^{-1} or atan):
- Calculate the ratio of y to x: (-60.0) / (-38.0).
- Take the arctan of this ratio: arctan(60.0 / 38.0).
- The angle is found to be approximately 57.5°.
However, because both components are negative, the vector lies in the third quadrant, where the reference angle given by arctan must be adjusted to find the angle from the positive x-axis. We do this by adding 180° to the reference angle (because angles in the third quadrant are between 180° and 270°).
Therefore, the angle is approximately 57.5° + 180° = 237.5°.
In summary, the vector has a magnitude of 70.71 units and a direction of approximately 237.5°.