Finding the exact angle between two initial vectors typically requires more information than just their resultant vector. However, if your goal is to determine the direction of the resultant vector itself relative to an axis (like the positive x-axis), the components of the resultant vector are exactly what you need.
Based on the provided information, we can describe how to find the angle that the resultant vector makes with the positive x-axis, using its x and y components.
The resultant vector is the sum of two or more vectors. If you know the components of the resultant vector (let's call them x and y), you can find its angle ($\theta$) relative to the positive x-axis using the inverse tangent function (arctan or tan⁻¹).
Finding the Angle of the Resultant Vector
The reference provides the method to calculate the angle of the resultant vector relative to the positive x-axis. This angle is determined by the ratio of the resultant vector's y-component to its x-component.
Here's how to find the angle ($\theta$) of the resultant vector from its components:
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Calculate the ratio of the y-component to the x-component: This ratio is $\frac{y}{x}$. Note that
yhere represents the sum of the y-components of the original vectors ($\Sigma y_i$) andxrepresents the sum of the x-components ($\Sigma x_i$). -
Use the inverse tangent function: The base angle is found using $\tan^{-1}(\frac{y}{x})$.
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Determine the correct quadrant for the angle: The inverse tangent function usually returns an angle between -90° and +90°. You need to adjust this angle based on the signs of the x and y components to get the correct direction (0° to 360°).
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Case 1: When the x-component (x) of the resultant is positive:
The angle $\theta$ is given directly by:
θ = tan⁻¹ ( y / x )
(This angle will be in Quadrant I if y is positive, and Quadrant IV if y is negative). -
Case 2: When the x-component (x) of the resultant is negative:
The angle $\theta$ needs to be adjusted by 180° to place it in the correct quadrant (Quadrant II or III).
The angle $\theta$ is given by:
θ = tan⁻¹ ( y / x ) ± 180°
(Use +180° if y is positive (Quadrant II), and -180° or +180° if y is negative (Quadrant III). Adding or subtracting 180° works because tan has a period of 180°. Using +180° generally places the angle correctly between 0° and 360° when x is negative).
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In either case, y and x are equal to the ∑yᵢ and ∑xᵢ from the prior formula (meaning, the sum of the individual y-components and x-components of the original vectors).
Example: Finding the Resultant Vector's Angle
Suppose the resultant vector has an x-component of $x = -3$ and a y-component of $y = 4$.
- Ratio: $\frac{y}{x} = \frac{4}{-3} \approx -1.333$
- Base angle: $\tan^{-1}(-1.333) \approx -53.13°$
- Adjust for quadrant: Since the x-component (-3) is negative, we are in Case 2.
$\theta = -53.13° + 180° = 126.87°$.
(The resultant vector with components (-3, 4) is in the second quadrant, and an angle of approximately 126.87° is correct).
If the resultant vector had components $x = 3$ and $y = -4$:
- Ratio: $\frac{y}{x} = \frac{-4}{3} \approx -1.333$
- Base angle: $\tan^{-1}(-1.333) \approx -53.13°$
- Adjust for quadrant: Since the x-component (3) is positive, we are in Case 1.
$\theta = -53.13°$. This angle is typically expressed as a positive angle between 0° and 360° by adding 360°: $-53.13° + 360° = 306.87°$.
(The resultant vector with components (3, -4) is in the fourth quadrant, and an angle of approximately 306.87° is correct).
Summary Table
| Resultant Vector Components (x, y) | Condition on x | Formula for Angle θ (relative to +x-axis) | Notes |
|---|---|---|---|
| (x, y) | x > 0 | θ = tan⁻¹ (y / x) | Angle in Quadrant I (y>0) or IV (y<0) |
| (x, y) | x < 0 | θ = tan⁻¹ (y / x) + 180° | Angle in Quadrant II (y>0) or III (y<0) |
| (0, y) | x = 0, y > 0 | θ = 90° | Vector points along positive y-axis |
| (0, y) | x = 0, y < 0 | θ = 270° (or -90°) | Vector points along negative y-axis |
| (x, 0) | x > 0, y = 0 | θ = 0° | Vector points along positive x-axis |
| (x, 0) | x < 0, y = 0 | θ = 180° | Vector points along negative x-axis |
| (0, 0) | x = 0, y = 0 | Undefined | Zero vector has no specific direction |
This method allows you to find the direction of the resultant vector itself based on its components, as described in the reference. It does not directly provide the angle between the two original vectors that were added to get this resultant.
