To find the magnitude of the resultant vector when adding vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, we need to first resolve each vector into its components along the x-axis and y-axis. Then, we sum the components to find the total x and y components of the resultant vector.

Step 1: Resolve each vector into its components

1. Vector $\mathbf{a}$ (25 N at 0°):

   • $a_x = 25 \cos(0^\circ) = 25 \times 1 = 25 \, \text{N}$
   • $a_y = 25 \sin(0^\circ) = 25 \times 0 = 0 \, \text{N}$

2. Vector $\mathbf{b}$ (40 N at 180°):

   • $b_x = 40 \cos(180^\circ) = 40 \times (-1) = -40 \, \text{N}$
   • $b_y = 40 \sin(180^\circ) = 40 \times 0 = 0 \, \text{N}$

3. Vector $\mathbf{c}$ (10 N at 90°):

   • $c_x = 10 \cos(90^\circ) = 10 \times 0 = 0 \, \text{N}$
   • $c_y = 10 \sin(90^\circ) = 10 \times 1 = 10 \, \text{N}$

Step 2: Find the resultant components

Now, we add up the components along the x and y axes.

• Total x-component:
  $R_x = a_x + b_x + c_x = 25 + (-40) + 0 = -15 \, \text{N}$

• Total y-component:
  $R_y = a_y + b_y + c_y = 0 + 0 + 10 = 10 \, \text{N}$

Step 3: Calculate the magnitude of the resultant vector

The magnitude of the resultant vector $\mathbf{R}$ is given by:

$|R| = \sqrt{R_x^2 + R_y^2}$

Substituting the values:

$|R| = \sqrt{(-15)^2 + 10^2} = \sqrt{225 + 100} = \sqrt{325} \approx 18.03 \, \text{N}$

Final Answer:

The magnitude of the resultant vector is approximately 18.03 N.

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Would you like a more detailed breakdown of the calculation, or do you have any specific questions about the steps? Here are some related questions you might find interesting:

1. How would the direction of the resultant vector be calculated?
2. What happens if one of the vectors is at an angle other than the standard 0°, 90°, or 180°?
3. How do vector components affect the overall magnitude of the resultant vector?
4. What is the difference between adding vectors in polar form and rectangular form?
5. How would you calculate the angle of the resultant vector from the x-axis?

Tip: Always check the direction of your vectors (angles) carefully when resolving them into components, as errors in sign can lead to incorrect results.