Answer :
To determine the angle above the x-axis, also known as "north of east," for a vector given by its components (15 m east and 8 m north), follow these steps:
1. Understand the Components: The vector has an x-component of 15 meters (east) and a y-component of 8 meters (north).
2. Use the Arctangent Function: The angle θ can be found using the arctangent function (tan^(-1)) of the ratio of the y-component to the x-component. This is because the tangent of the angle θ is equal to the opposite side (y-component) over the adjacent side (x-component).
[tex]\[ \theta = \tan^{-1} \left( \frac{y \text{ component}}{x \text{ component}} \right) \][/tex]
Plugging in the given values:
[tex]\[ \theta = \tan^{-1} \left( \frac{8}{15} \right) \][/tex]
3. Calculate the Angle in Radians: Evaluating this expression gives the angle in radians.
[tex]\[ \theta \approx 0.4899573262537283 \text{ radians} \][/tex]
4. Convert Radians to Degrees: To convert the angle from radians to degrees, use the conversion factor [tex]\(1 \text{ radian} \approx 57.2958 \text{ degrees}\)[/tex].
[tex]\[ \theta \text{ in degrees} = 0.4899573262537283 \text{ radians} \times 57.2958 \text{ degrees/radian} \approx 28.072 \text{ degrees} \][/tex]
5. Round to Three Significant Figures: The final step is to express the angle in degrees to three significant figures.
[tex]\[ \theta \approx 28.072 \text{ degrees} \][/tex]
Therefore, the angle above the x-axis (north of east) for the vector with components (15 m, 8 m) is approximately [tex]\(28.072\)[/tex] degrees.
1. Understand the Components: The vector has an x-component of 15 meters (east) and a y-component of 8 meters (north).
2. Use the Arctangent Function: The angle θ can be found using the arctangent function (tan^(-1)) of the ratio of the y-component to the x-component. This is because the tangent of the angle θ is equal to the opposite side (y-component) over the adjacent side (x-component).
[tex]\[ \theta = \tan^{-1} \left( \frac{y \text{ component}}{x \text{ component}} \right) \][/tex]
Plugging in the given values:
[tex]\[ \theta = \tan^{-1} \left( \frac{8}{15} \right) \][/tex]
3. Calculate the Angle in Radians: Evaluating this expression gives the angle in radians.
[tex]\[ \theta \approx 0.4899573262537283 \text{ radians} \][/tex]
4. Convert Radians to Degrees: To convert the angle from radians to degrees, use the conversion factor [tex]\(1 \text{ radian} \approx 57.2958 \text{ degrees}\)[/tex].
[tex]\[ \theta \text{ in degrees} = 0.4899573262537283 \text{ radians} \times 57.2958 \text{ degrees/radian} \approx 28.072 \text{ degrees} \][/tex]
5. Round to Three Significant Figures: The final step is to express the angle in degrees to three significant figures.
[tex]\[ \theta \approx 28.072 \text{ degrees} \][/tex]
Therefore, the angle above the x-axis (north of east) for the vector with components (15 m, 8 m) is approximately [tex]\(28.072\)[/tex] degrees.
