Certainly! Let's determine the direction (or angle) of the vector given its components.
The vector has the following components:
- [tex]$x$[/tex]-component: [tex]$-1.55$[/tex] meters
- [tex]$y$[/tex]-component: [tex]$3.22$[/tex] meters
To find the direction of the vector, we need to calculate the angle it makes with the positive x-axis. This angle can be found using the arctangent function, which gives us the angle whose tangent is the ratio of the [tex]$y$[/tex]-component to the [tex]$x$[/tex]-component.
Mathematically, the direction angle [tex]$\theta$[/tex] can be determined using:
[tex]\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\][/tex]
However, since we are specifically dealing with both positive and negative components, and to ensure the correct quadrant for the angle, we use the [tex]$\text{atan2}$[/tex] function. This function computes the arctangent of the ratio while taking into account the signs of both components, thereby giving us the correct angle in the coordinate plane.
Using the given components:
[tex]\[
\theta = \text{atan2}(3.22, -1.55)
\][/tex]
After computing the [tex]$\text{atan2}$[/tex] function and converting the result from radians to degrees, we obtain the direction of the vector. The result for this specific example is:
[tex]\[
\theta \approx 115.7046^\circ
\][/tex]
Hence, the direction of the vector is:
[tex]\[
115.7046^\circ
\][/tex]
