To find the direction of the resultant vector when adding vectors $a + b + c + d$, we need to break each vector into its horizontal (x) and vertical (y) components and then sum them up.

Step 1: Break each vector into components

1. Vector $a$ (25 N at 0 degrees):

   • Horizontal component (x): $a_x = 25 \cos(0^\circ) = 25 \, \text{N}$
   • Vertical component (y): $a_y = 25 \sin(0^\circ) = 0 \, \text{N}$

2. Vector $b$ (35 N at 180 degrees):

   • Horizontal component (x): $b_x = 35 \cos(180^\circ) = -35 \, \text{N}$
   • Vertical component (y): $b_y = 35 \sin(180^\circ) = 0 \, \text{N}$

3. Vector $c$ (50 N at 270 degrees):

   • Horizontal component (x): $c_x = 50 \cos(270^\circ) = 0 \, \text{N}$
   • Vertical component (y): $c_y = 50 \sin(270^\circ) = -50 \, \text{N}$

4. Vector $d$ (30 N at 90 degrees):

   • Horizontal component (x): $d_x = 30 \cos(90^\circ) = 0 \, \text{N}$
   • Vertical component (y): $d_y = 30 \sin(90^\circ) = 30 \, \text{N}$

Step 2: Sum the components

• Total horizontal component (x): $R_x = a_x + b_x + c_x + d_x = 25 + (-35) + 0 + 0 = -10 \, \text{N}$

• Total vertical component (y): $R_y = a_y + b_y + c_y + d_y = 0 + 0 + (-50) + 30 = -20 \, \text{N}$

Step 3: Calculate the magnitude of the resultant vector

$R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-10)^2 + (-20)^2} = \sqrt{100 + 400} = \sqrt{500} \approx 22.36 \, \text{N}$

Step 4: Calculate the direction (angle) of the resultant vector

The angle $\theta$ is given by: $\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{-20}{-10}\right) = \tan^{-1}(2)$ $\theta \approx 63.43^\circ$

Since both $R_x$ and $R_y$ are negative, the vector lies in the third quadrant, so the angle is measured from the negative x-axis: $\theta = 180^\circ + 63.43^\circ = 243.43^\circ$

Result:

The direction of the resultant vector is approximately 243.43 degrees.

Would you like more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How would the direction change if vector $a$ was at 90 degrees instead of 0 degrees?
2. What is the resultant magnitude if one of the vectors had a different force value?
3. How do we add vectors in three-dimensional space?
4. How can we calculate the resultant vector if the vectors are not given at cardinal directions?
5. What role does the angle of the vector play in determining the resultant?

Tip: Always break vectors into components (x and y) when solving vector addition problems.