What is the direction of the resultant vector when adding (a+b+c+d), when a is 35 N at 0 degrees, b is 45 N at 180 degrees, c is 40 N at 270 degrees, and d is 20 N at 90 degrees?

To find the direction of the resultant vector, we need to break down each vector into its horizontal (x) and vertical (y) components, add the components, and then use the resultant components to calculate the angle.

### Step 1: Break the vectors into components

1. **Vector a** (35 N at 0 degrees):
   - \( x_a = 35 \, \text{N} \times \cos(0^\circ) = 35 \, \text{N} \)
   - \( y_a = 35 \, \text{N} \times \sin(0^\circ) = 0 \, \text{N} \)

2. **Vector b** (45 N at 180 degrees):
   - \( x_b = 45 \, \text{N} \times \cos(180^\circ) = -45 \, \text{N} \)
   - \( y_b = 45 \, \text{N} \times \sin(180^\circ) = 0 \, \text{N} \)

3. **Vector c** (40 N at 270 degrees):
   - \( x_c = 40 \, \text{N} \times \cos(270^\circ) = 0 \, \text{N} \)
   - \( y_c = 40 \, \text{N} \times \sin(270^\circ) = -40 \, \text{N} \)

4. **Vector d** (20 N at 90 degrees):
   - \( x_d = 20 \, \text{N} \times \cos(90^\circ) = 0 \, \text{N} \)
   - \( y_d = 20 \, \text{N} \times \sin(90^\circ) = 20 \, \text{N} \)

### Step 2: Add the components

- \( \text{Resultant } x \text{ component} = x_a + x_b + x_c + x_d = 35 \, \text{N} + (-45 \, \text{N}) + 0 + 0 = -10 \, \text{N} \)
- \( \text{Resultant } y \text{ component} = y_a + y_b + y_c + y_d = 0 + 0 + (-40 \, \text{N}) + 20 \, \text{N} = -20 \, \text{N} \)

### Step 3: Calculate the direction of the resultant vector

The direction (angle \( \theta \)) of the resultant vector can be found using the formula:
\[
\theta = \tan^{-1} \left( \frac{y_{\text{resultant}}}{x_{\text{resultant}}} \right)
\]
Substituting the values:
\[
\theta = \tan^{-1} \left( \frac{-20}{-10} \right) = \tan^{-1}(2) = 63.4^\circ
\]

Since both the \(x\) and \(y\) components are negative, the angle lies in the third quadrant. Thus, we add 180° to the angle:

\[
\theta = 63.4^\circ + 180^\circ = 243.4^\circ
\]

### Result
The direction of the resultant vector is approximately **243.4°**.

Do you want details or have any questions?

### Related Questions:
1. What is the magnitude of the resultant vector for the same problem?
2. How would the resultant vector change if vector \(d\) had a different magnitude?
3. How do we resolve vectors when the angles are not multiples of 90 degrees?
4. How can we calculate the angle if the resultant is in different quadrants?
5. Can this method be used for three-dimensional vector addition?

### Tip:
When resolving vectors, always check the quadrant based on the signs of the components before determining the final angle.

Find the direction of the resultant vector when adding vectors with magnitudes of 35 N, 45 N, 40 N, and 20 N at 0°, 180°, 270°, and 90° respectively. The solution covers vector addition using components, trigonometric identities, and the inverse tangent function, suitable for Grades 10-12 students.

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