To determine the angle between the x-axis and the force vector [tex]\( \mathbf{F} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \)[/tex], follow these steps:
1. Identify the force vector:
Given [tex]\( \mathbf{F} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \)[/tex], we can represent it in vector form as [tex]\( \mathbf{F} = [2, 3, 4] \)[/tex].
2. Identify the unit vector along the x-axis:
The unit vector along the x-axis is [tex]\( \mathbf{i} \)[/tex], which can be written as [tex]\( \mathbf{i} = [1, 0, 0] \)[/tex].
3. Calculate the magnitudes of the vectors:
- The magnitude of [tex]\( \mathbf{F} \)[/tex] is given by:
[tex]\[
|\mathbf{F}| = \sqrt{2^2 + 3^2 + 4^2} = 5.385164807134504
\][/tex]
- The magnitude of the x-axis unit vector [tex]\( \mathbf{i} \)[/tex] is:
[tex]\[
|\mathbf{i}| = \sqrt{1^2 + 0^2 + 0^2} = 1.0
\][/tex]
4. Calculate the dot product between [tex]\( \mathbf{F} \)[/tex] and [tex]\( \mathbf{i} \)[/tex]:
The dot product [tex]\( \mathbf{F} \cdot \mathbf{i} \)[/tex] is:
[tex]\[
2 \cdot 1 + 3 \cdot 0 + 4 \cdot 0 = 2
\][/tex]
5. Calculate the angle:
Using the dot product formula to find the angle [tex]\( \theta \)[/tex] between the two vectors:
[tex]\[
\cos(\theta) = \frac{\mathbf{F} \cdot \mathbf{i}}{|\mathbf{F}| \cdot |\mathbf{i}|}
\][/tex]
Substituting in the known values:
[tex]\[
\cos(\theta) = \frac{2}{5.385164807134504 \cdot 1.0} = \frac{2}{5.385164807134504}
\][/tex]
Thus, solving for [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \arccos\left( \frac{2}{5.385164807134504} \right) = 1.1902899496825317 \, \text{radians}
\][/tex]
6. Convert the angle from radians to degrees:
Using the conversion factor [tex]\( 180^\circ / \pi \)[/tex]:
[tex]\[
\theta \, \text{(degrees)} = 1.1902899496825317 \, \text{radians} \times \frac{180^\circ}{\pi} = 68.19859051364818^\circ
\][/tex]
Therefore, the angle between the x-axis and the force vector [tex]\( \mathbf{F} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \)[/tex] is approximately [tex]\( 68.20^\circ \)[/tex].
